The Perturbation of Hamiltonian Systems with a Non-Abelian Symmetry Citation Blaom, Anthony David (1998) The Perturbation of Hamiltonian Systems with a Non-Abelian Symmetry. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/refg-y469. https://resolver.caltech.edu/CaltechTHESIS:07252025-175651501 Abstract The perturbation theory of non-commutatively integrable systems is revisited from the point of view of non-Abelian symmetry groups. Using a coordinate system intrinsic to the geometry of the symmetry, we generalize well-known estimates of Nekhoroshev (1977) in a class of systems having almost G-invariant Hamiltonians. These estimates are shown to have a natural interpretation in terms of momentum maps and co-adjoint orbits. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Marsden, Jerrold E. (advisor) Luxemburg, W. A. J. (co-advisor) Thesis Committee: Marsden, Jerrold E. (chair) Gabai, David Ratiu, Tudor Simon, Barry M. Defense Date: 19 March 1998 Record Number: CaltechTHESIS:07252025-175651501 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07252025-175651501 DOI: 10.7907/refg-y469 ORCID: Author ORCID Blaom, Anthony David 0000-0001-6689-886X Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17558 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 25 Jul 2025 19:15 Last Modified: 25 Jul 2025 19:35 Thesis Files PDF - Final Version See Usage Policy. 52MB Repository Staff Only: item control page

The perturbation of Hamiltonian systems with a non-Abelian symmetry Thesis by Anthony D. Blaom In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1998 (Defended March 19 , 1998) ii Copyright © 1998 Anthony D. Blaom All Rights Reserved To my father, Anthony E. Perry, for inspiration. Abstract The perturbation theory of non-commutatively integrable systems is revisited from the point of view of non-Abelian symmetry groups. Using a coordinate system intrinsic to the geometry of the -symmetry, we generalize well-known estimates of Nekhoroshev (1977) in a class of systems having almost G-invariant Hamiltonians. These estimates are shown to have a natural interpretation in terms of momentum maps and co-adjoint orbits. iv Contents Abstract iv Overture X Two simple Hamiltonian systems X The complexity of perturbed motions XI Nekhoroshev's theorem XU Non-commutative integrability xv The symmetry point of view xvii Action-group coordinates xviii Nekhoroshev estimates in the non-commutative case xxi Thesis outline xxi Acknowledgments xxm Introduction 1 Non-commutative integrability 1 Integrability through symmetry 3 Generalized Nekhoroshev estimates 3 On the construction of action-group coordinates 5 Part 1. Dynamics 7 Outline of Part 1 Chapter 1. 8 Preliminaries 9 Wey! chambers 9 Wey! chambers in g* 11 V CONTENTS Chapter 2. vi Action-group coordinates 12 The model space and its symplectic structure 12 Hamiltonian vector fields in action-group coordinates 14 Equations of motion in action-group coordinates 15 The Poisson bracket in action-group coordinates 18 Chapter 3. On the existence of action-group coordinates 20 System symmetry breaking 21 Orbit type conditions 21 Integrability 24 The existence theorem 26 Historical remarks 27 Naive averaging 30 The unperturbed dynamics 30 The averaged dynamics 33 Chapter 4. Chapter 5. An abstract formulation of Nekhoroshev 's theorem 38 Notation 39 Preliminary notions 39 The abstract set-up 40 Fundamental assumptions 41 Nekhoroshev's theorem 43 Chapter 6. Applying the general theorem to action-group coordinates 46 The local complexification of G 47 Existence of a period lattice (Assumption A) 49 Estimating c1 50 Bounds on Ap 51 Bounds on directional derivatives 54 Times of validity of Hamiltonian flows (Assumption B) 56 CONTENTS Inequalities for Poisson brackets (Assumption C) Chapter 7. Part 2. Nekhoroshev-type estimates for momentum maps Geo metry Outline of Part 2 Chapter 8. On Hamiltonian G-spaces with regular momenta vii 59 62 65 66 68 Preliminaries 68 On the geometry of the co-adjoint action 70 The fibering of P by Hamiltonian T-spaces 71 The symplectic connection 73 A simple example: Geodesic motions on the sphere 74 Reduction to the Abelian case 75 The rr-invariant T-action 76 Chapter 9. Action-group coordinates as a symplectic cross section 79 The symplectic cross section of T*G 79 Action-group coordinates as a Hamiltonian G-space 82 Hamiltonian vector fields in action-group coordinates 82 The Poisson bracket in action-group coordinates 84 Chapter 10. Constructing action-group coordinates 86 The non-Abelian case 86 The Abelian case 91 Chapter 11. The axisymmetric Euler-Poinsot rigid body 97 Problem prescription 97 Existence 99 The sym plectic cross section 101 Explicit action-group coordinates 102 Chapter 12. Converting dynamic integrability to geometric integrability 104 CONTENTS viii Setup 105 The 3 1 extension 106 Aside: On computing phases 109 Aside: Relative equilibria and 'limiting phases' 110 Proof of the 5 1 Extension Theorem 112 The non-Abelian case 116 Chapter 13. Concluding remarks 117 Generalizations 117 Fourier series 118 Appendix A. Proof of the Nekhoroshev-Lochak Theorem (abstract form) 119 Averaging over periodic orbits 119 Lie transforms and the Iterative Lemma 120 The Hamiltonian in normal form 125 Moving coordinate systems and reduction to the zero frequency case 127 Nekhoroshev estimates in the neighborhood of the periodic orbits 132 Estimates for an arbitrary initial condition 135 Appendix B. Proof that W is a slice 140 Appendix C. Proof of the Extension Lemma 142 Appendix D. An application of the 3 1 Extension Theorem: The axisymmetric EulerPoinsot rigid body revisited 145 The symplectic cross section 146 The axisymmetric case 149 Constructing the 3 1 extension in the symplectic cross section 151 Appendix E. The Symplectic Leaf Correspondence Theorem 153 Foliations with singularities 153 Symplectic leaves and dual pairs 154 CONTENTS IX The Symplectic Leaf Correspondence Theorem 155 Details 157 Bibliography 161 Overture A detailed exposition of the results to be expounded in this dissertation will be offered in the Introduction. While some of these results are sketched here, the main task at hand is to describe, informally and in plain language, the questions and problems motivating our work, as well as the ideas and point of view leading us ultimately to answers. It will be illustrative to keep two examples in mind that serve as good prototypes for the larger class of systems that will be studied later. Two simple Hamiltonian systems Example A. Consider the (classical) motion of a point mass m moving through ordinary three-dimensional space, free of external forces including gravity. The linear momentum J1in of the mass is conserved, so that the motion is in a straight line at constant speed. Example B . Alternatively, suppose that our point mass is constrained to move on the surface of a smooth sphere, the only external force being the normal force necessary to maintain the constraint. In this case the angular momentum J ang is conserved and the motion is along great circles of the sphere at constant speed . / A J,;" Jang l:. ·'. .'m , X ,, ,, B / /'r' \ \ r:- THE COMPLEXITY OF PERTURBED MOTIONS XI Both Examples A and B possess a conservation law. The conservation laws can be 'explained' by the existence of symmetries in the underlying equations. This is the content of a well-known theorem of Noether (see, e.g., Marsden (1992)). Example A clearly possesses a three-dimensional translational symmetry, while Example B possesses a rotational symmetry. Suppose these symmetries are broken. For instance, suppose that in Example A we add a weak spatially periodic potential, and in Example B we introduce a small aspherical distortion. What is the fate of the above conservation laws? The complexity of perturbed motions In the absence of perturbations, Examples A and B are both integrable systems, i.e., systems whose solutions can be written down in an explicit or 'closed' form . Unfortunately, integrability is the exception rather than the rule, and generic perturbations to integrable systems create extraordinarily complicated behavior. Numerically such behavior is manifest in spectacular fractal-like phase portraits (see, e.g., Fig. 1) , broadband Fourier spectra (Noid, Koszykowski and Marens, 1977), exponential divergence of nearby trajectories (Benettin and Galgani, 1979) and computations of KS entropy (Chirikov, 1979). On the analytical side, we have the existence of homo- and heteroclinic tangles, giving rise to horseshoe maps and chaotic dynamics on Cantor sets (Moser, 1973; Smale, 1967). According to Poincare's nonexistence theorem (see, e.g., Benettin, Ferrari, Galgani and Giorgilli (1982)) conserved quantities in an integrable system are irrevocably destroyed by generic perturbations. Despite this and our preceding remarks, there are two important theorems of canonical perturbation theory that establish , under appropriate hypotheses, some kind of 'stability' for the conserved quantities. These are the Kolmogorov-Arnold-Moser (or KAM) theorem (Kolmogorov, 1954; Arnold, 1963; Moser, 1962) and Nekhoroshev's theorem (Nekhoroshev, 1977). For generic perturbations, the KAM theorem does not directly apply to Example B (or to other examples like it) , and we discuss it no further 1 . Nekhoroshev's 1 The interested reader is referred to, e.g., Arnold, Kozlov and Neishtadt (1988). NEKHOROSHEV'S THEOREM xii 0.85 0.75 X 0.65 0.55 0.45 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 dx/dt FIGURE 1. A phase portrait made by iterating selected points under a return map for the restricted three-body problem with Jacobi constant C = 3.1 and with µ = 0.001, the approximate mass ratio between Jupiter and the Sun. theorem can be applied to both Examples A and B, but not without significant differences that we now elucidate. Nekhoroshev's theorem Nekhoroshev's theorem applies to systems admitting action-angle coordinates. Nekhoroshev also stated his result in terms of a generalization called partial action-angle coordinates, but his argument in the more general setting was incomplete. A complete argument using partial action-angle coordinates was offered only recently (Fasso, 1995). This will be relevant to our discussion of Example B, but not in Example A, which we describe first . NEKHOROSHEV'S THEOREM xiii Action-angle coordinates in Example A are easily constructed. Suppose as before, that attention is restricted to spatially periodic perturbations, and let L be a 'lattice of periodicity.' For simplicity, assume that L has an orthogonal basis, and that the corresponding periods of L are the same in each direction. Non-dimensionalize all lengths by this period, and all masses by the mass of m. Modulo the lattice L, the position of m is determined by three angles q1 , q2 , q3 . The associated 'actions' are the components P1, pz, p3 of the momentum vector J1in· What makes these coordinates action-angle coordinates is that: (i) they put the equations of motion into Hamiltonian form, . 8H q· - J - . 8pj 8H Pj = - ~ , uqj where the appropriate Hamiltonian His in this case the total energy of the system, and (ii) the unperturbed value Ho of the Hamiltonian (i.e., the kinetic energy) depends only on the action variables: In particular, the unperturbed motions of a system in action-angle coordinates are quasiperiodic: where 0,j = 8h/ 8pj (p(O)). In our example, we have 0,j = Pj (0), and the constancy of the action variables corresponds to the conservation of momentum J1in· We are now ready to apply Nekhoroshev's result: NEKHOROSHEV'S THEOREM xiv Nekhoroshev's Theorem. (Nekhoroshev (1977) 2 ) Consider a Hamiltonian system in action-angle coordinates qi, ... , qn, Pi, ... , Pn and consider perturbed Hamiltonians of the form H(q,p) = h(p) + EF(q,p) . Restrict attention to values of the action vector p = (Pi, ... , Pn) lying in some finite closed ball B, and initial conditions p(O) lying in the interior of B. Assume h and F are realanalytic, and that the level sets in B of the unperturbed Hamiltonian h are strictly convex. Then there exist positive constants a, b, c, to and r 0 such that for all sufficiently small E ~ 0 all solutions of the perturbed system satisfy the exponential estimate lp(t) - p(O)I ~ roi . In Example A we have p = J1in, so that supposing that the perturbation varies proportionally with some parameter E, Nekhoroshev's theorem predicts that order i 'drifts' in the momentum J1in of the mass m take times on the order of exp(cca). If Eis small, such times can be very large indeed. To demonstrate just how large 'very large' can be in practice, we will relate a vivid example drawn from celestial mechanics. In Giorgilli and Skokos (1997) Nekhoroshev type estimates are derived for Trojan asteroids in a restricted three-body model of the Jupiter-Sun system. In an appropriate rotating system of coordinates, the model has an equilibrium solution known as the Lagrange point 1 4 . It is shown that an asteroid beginning at rest (in the rotating coordinates) and lying within 0.127 D of L 4 , where Dis the distance from Jupiter to the Sun, will not increase that distance by a factor exceeding 1.05 before the universe has doubled its current estimated age! There are, moreover, at least four such Trojan asteroids presently known. 2 For this formulation see, e.g., Lochak (1992). In Nekhoroshev's original statement, the convexity condition is replaced by a 'steepness' condition. This is a weaker (indeed C 00 -generic) condition we will not attempt to consider here or elsewhere. NON-COMMUTATIVE INTEGRABILITY xv Non-commutative integrability Before turning to Example B, let us summarize some key observations about Example A, which are typical of systems admitting action-angle coordinates: Al. Unperturbed motions in Example A (which has three degrees of freedom) are quasiperiodic with three independently controllable 3 frequencies. A2. The conserved quantity J1in is a vector with three components. A3. The underlying translational symmetry is Abelian, meaning that the net effect of two successive translations is independent of the order in which they are applied. We add one final observation which is less obvious but nevertheless important: A4. The Poisson bracket (see below) {Ji, Jj} of any pair of components J 1 , h, h of hn vanishes. By definition, the Poisson bracket {J, h} of two functions f and his computed by differentiating f along solution curves of the system obtained by taking as Hamiltonian the function h. If we are to apply Nekhoroshev's theorem as above to Example B, then we shall first need to construct action-angle coordinates. But at odds with this requirement is the disturbing fact that analogues of the observations Al-A4 follow an altogether different pattern in Example B: Bl. Unperturbed motions in Example B (which has two degrees of freedom) are periodic, i.e., have only one associated frequency. B2. The conserved quantity Jang has three components, i.e., one more component than the system has degrees of freedom. B3. The underlying rotational symmetry is non-Abelian, since the net effect of applying successive rotations in three-space depends in general on the order in which these rotations are applied. 3 Controllable by varying the initial conditions Pi (0). NON-COMMUTATIVE INTEGRABILITY xvi B4. The Poisson brackets of components J1, Jz, Jz of Jang satisfy the cyclic conditions What is remarkable is that (local) action-angle coordinates can be constructed in Example B although, as one might imagine in view of the incongruence of the properties listed above, the construction is substantially more complicated than was the case in Example A. We do not attempt to describe this construction here. At any rate, the unperturbed Hamiltonian H 0 (q,p) = h(p) in action-angle coordinates fails to be convex, so that Nekhoroshev's theorem above fails to apply. Integrable systems whose conserved quantities satisfy nontrivial Poisson bracket relations, as in B4, are known as non-commutatively integrable systems. In work proceeding the research on exponential estimates, Nekhoroshev (1972) determined that a large class of non-commutatively integrable systems admit a generalization of action-angle coordinates known as partial action-angle coordinates, which are in some sense more natural. Partial action-angle coordinates consist of two sets of variables: The first set consists of k angles q1, ... , qk and k conjugate actions p 1, ... , Pk. The second set consists of (n - k) variables x1, ... , Xn-k and (n - k) conjugate variables Y1, ... , Yn-k, which are all ordinary (i.e., nonangular) coordinate functions. Here n denotes the total number of degrees of freedom. These coordinates, like conventional action-angle coordinates, are canonical in the sense that they put the equations of motion into Hamiltonian form: . fJH fJpj q· - J - . fJH fJqj p· - - J - . fJH x· - J - fJyj . 8H y· - - J ox· J In partial action-angle coordinates, a non-commutatively integrable Hamiltonian depends only on the Pj variables . THE SYMMETRY POINT OF VIEW xvii In Nekhoroshev's work on exponential estimates, he realized that most of his arguments carry over to the non-commutatively integrable case if one uses partial action-angle coordinates, and assumes convexity of the unperturbed Hamiltonian with respect to the Pj variables only. Even in Example B, however (the simplest example of a non-commutatively integrable system imaginable!), constructing partial action-angle coordinates is not trivial. Furthermore, one cannot construct partial action-angle coordinates globally in Example B without coordinate singularities. This is true even if we remove points in phase space corresponding to trivial motions (the mass mat rest), which constitute a 'natural' singularity of the problem. Difficulties posed by coordinate singularities are well-known in celestial mechanics; see, e.g., Coffey, Deprit and Miller (1986) and the references contained therein. Another problem with Nekhoroshev's generalized setting, as Fasso (1995) has pointed out, is that 'fast' (i.e., order E) motions in the Xj, Yj variables take solutions to the perturbed problem in a non-commutatively integrable system out of locally defined partial action-angle coordinate charts, before the exponential estimates on the Pj variables can be rigorously established. Fasso overcomes this problem by showing how to make intrinsic sense of 'normal forms' for the perturbed Hamiltonian, although to express and compute Nekhoroshev type estimates still requires one to fix an atlas of partial action-angle coordinate charts. The extent to which these estimates depend on the choice of atlas is an open problem (Fasso, 1995, Appendix C). The symmetry point of view The reason for the difficulties presented by action-angle or partial action-angle coordinates in non-commutatively integrable systems is that the canonical nature of such coordinates is at odds with the intrinsic non-Abelian symmetry underlying many such systems. In this thesis we take the point of view that the geometry of an underlying symmetry, as well as its corresponding conservation law, should be built into whatever geometric framework is employed to carry out an analysis of perturbations. This viewpoint is to take precedence over previous requirements that one work exclusively with canonical coordinate systems. ACTION-GROUP COORDINATES xviii Rather, one should endeavor to understand how non-canonical contributions enter the equations of motion, when these are viewed in a coordinate system intrinsic to the non-Abelian symmetry. With this kind of understanding in hand, we seek to geometrize and generalize the commutative version of Nekhoroshev's theorem given above, in such a way that its application to systems with a perturbed non-Abelian symmetry becomes transparent. In particular, this generalization should yield estimates that apply immediately to the conservation law intrinsically associated with such a symmetry. Action-group coordinates The simplest coordinate system of the kind we advocate here is one that we shall refer to as action-group coordinates. In Example A, and other systems whose integrability arises from the existence of an appropriate Abelian symmetry, action-group coordinates correspond to conventional action-angle coordinates. A discussion of action-group coordinates in general is postponed to Chap. 2. We preview these coordinates here in the special case of Example B, which we shall revisit throughout our exposition of the general theory. A more sophisticated demonstration of this theory (Chap. 11) will be an application to the Euler-Poinsot rigid body. Nondimensionalize all lengths in Example B by the radius of the sphere, and all masses by that of m. The state of the system is determined by the position q and instantaneous linear momentum vector v of m. We view q and v and vectors in three-space, subject to the conditions llqll = 1 and v • q = 0 consistent with the physical constraints. Assume the unit sphere is centered on the origin of an inertial orthonormal frame with basis e 1 , e2 , e3 . It is convenient to express states (q, v) of m with respect to a reference state (q0 , v0 ), which we choose arbitrarily to be (e 1 , e2 ) (see Fig. 2). For an arbitrary state (q, v) there exists a 3 x 3 orientation preserving rotation matrix g and a number p ~ 0 such that q = gqo v = pgv0 ACTION-GROUP COORDINATES FIGURE XIX 2. The reference state (q 0 , v 0 ) and general state (q, v) of the mass min Example B. (The basis vectors e1 , e2 , e3 have been translated for clarity.) If we neglect trivial states (v = 0), then g and p are determined uniquely, and p is strictly positive. Whence g, p constitute 'coordinates,' in the sense that there is a one-to-one correspondence between states (q, v) and values for the pair (g, p). Notice that the rotational symmetry of the unperturbed system is completely transparent in the coordinates (g,p) as the unperturbed Hamiltonian H 0 (g,p) (i.e., the kinetic energy determined by the standard metric on the sphere 4 ) is independent of the 'symmetry' coordinate g; it depends only on the 'action' coordinate p, in the spirit of constructions of conventional action-angle coordinates: Ho(g, p) = h(p) 4 =21p 2 We realize perturbations in the form of aspherical distortions by perturbing this metric, not by per- turbing the sphere with which configurations q are identified. ACTION-GROUP COORDINATES The equations of motion take the form loH p092 loH !J=g -p 091 oH Bi . fJH p--- 893 ' where O -6 6 o -6 -6 6 o 6 and fJH ( ) - d H( te ) I Ogj g,p = dt ge J ,P t=D fJH d fJp (g,p) = dtH(g,p+ t)lt=O Roughly speaking, the non-canonical contributions to the equations are the terms l fJH l fJH In the unperturbed case, the equations reduce to p=O, which admit the general solution p(t) = p(O) g(t) = g(O) exp(tf!e3) , where Q = p(O). xx THESIS OUTLINE xxi Nekhoroshev estimates in the non-commutative case The angular momentum of min Example Bis given by Jang= q Xv= pg(qo X vo) = pge3 If one keeps p > 0 constant and runs through all possible values of the symmetry variable g, then the corresponding locus of Jang in three-space is a sphere of radius p. In fact these spheres are intrinsic geometric objects associated with the rotational symmetry of the problem known as co-adjoint orbits. These orbits (which are zero dimensional in the Abelian case) influence the perturbed motions, and in particular the fate of the conservation law jang = 0, in a fundamental way. Specifically, Nekhoroshev type estimates for a generic per- turbation can only be applied to 'drifts' in the momentum Jang in those directions transverse to the co-adjoint orbits (spheres). That is, Nekhoroshev estimates, under appropriate hypotheses, apply to the action variable p = IIJangll- Relatively fast motions (order E) can and do appear in the tangential directions, being projections onto three-space of fast dynamics in the g variable. These motions correspond to those pointed out by Fasso and discussed above. Unlike partial action-angle coordinate charts however, an action-group coordinate chart completely contains such motions. Thesis outline This concludes our informal introduction to the main themes of this dissertation. After a formal exposition of our results detailed in the Introduction, the thesis splits roughly into two parts: In Part 1, we concentrate on dynamics and analytical aspects, and in particular on the derivation of Nekhoroshev estimates in the framework of 'non-canonical' coordinate systems. In Part 2, we describe geometric constructions underlying the theory in detail. The reader primarily interested in geometric aspects is advised to read the Introduction and the first three chapters of Part 1, although one might get away with less. More detailed outlines preface each Part. Action-group coordinates are in fact just a special case of a more general notion of 'sy mmetry-intrinsic coordinates' known as Hamiltonian G-space normal forms. While the THESIS OUTLINE XXll efforts described here apply to some nontrivial systems, such as the Euler-Poinsot rigid body, they are really just the first step in a more extensive program based on such normal forms. We discuss this further in our Concluding Remarks (Chap. 13). Introduction 1 Part 1. Dynamics 8 Part 2. Geometry 66 Acknowledgments I thank my dear wife Thyra, and my Mum and Dad, for inestimable love and support. I thank Jerry Marsden for being an understanding advisor, who gives generously of his time and ideas. I also thank him for constant and patient support, and for allowing me freedom and independence wherever I sought it. For my close friend Muruhan Rathinam, I am grateful for sharing a love of music, wine and good food, and for initiating me in the Tea Ceremony. I also thank my other good friends Scott and Antonio for their constant companionship. I thank my colleagues Tudor Ratiu, Francesco Fasso and Pierre Lochak, for helpful feedback, and my former advisor Steve Wiggins for introducing me to perturbation theory. The author is also indebted to Alan Weinstein for pointing out the work of Marie, and that of Dazord and Delzant . Finally, I thank those many who helped me cope with chronic health problems, especially my wife Thyra, my parents, my advisor Jerry Marsden, Helen Hasenfeld, and the staff of the Office for Student Affairs. xxiii Introduction The usual starting point of Hamiltonian perturbation theory is the construction of a symplectic diffeomorphism between some open subset of the phase space P and an appropriate open subset of a 'model space' M, such that the unperturbed Hamiltonian is represented by a function on M whose integrability is readily apparent. When the integrability results from the existence of a sufficient number of Poisson-commuting integrals, and the joint level sets of these integrals are compact, the appropriate choice of model space is actionangle coordinates (M = 'llsn x Rn, n = ½dim P), whose existence is then guaranteed by the Arnold-Liouville theorem (Arnold et al., 1988) . Another model space that has appeared in recent practice (Fass<\ 1995) is a generalization introduced by Nekhoroshev (1972) known as partial (or generalized:) action-angle coordinates (M = 1'k x Rk x R 2 (n-k)). These coordinates are applicable when the integrability is 'non-commutative' (see below). In either case the model space is canonical in the sense that, up to a covering , it is Euclidean space equipped with its usual symplectic structure. Non-commutative integrability In geometric language, a Hamiltonian Ho, defined on some symplectic manifold (P,w), is integrable if Ho is constant on the leaves of a coisotropic and symplectically complete foliation F on P (Dazord and Delzant, 1987). A foliation F is coisotropic if its tangent distribution D contains its symplectic orthogonal Dw. It is symplectically complete if Dw is integrable as a distribution. In that case, the leaves of the corresponding foliation Fw are invariant under the flow of the Hamiltonian vector field XHo. If the leaves of Fw are compact, then they are in fact k dimensional tori (k = dim Fw = 2n-dim F) and Fw is an 'angular fibering' in the sense of Nekhoroshev (1972). In particular, NON-COMMUTATIVE INTEGRABILITY 2 in the case of commutative integrability (k = n) a neighborhood of each k-torus admits action-angle coordinates, while in the case of non-commutative integrability (k < n) a neighborhood of each such torus admits partial action-angle coordinates. In either case, the Hamiltonian Ho is represented by a function depending only on the k actions 'conjugate' to the angles representing the tori. This has the consequence that XH0 restricted to a given k-torus is conjugate to a linear vector field (i.e., is 'covered' by a linear vector field on IR k). Proofs of these statements and further details are given in Dazord and Delzant (1987) . Generally, in the non-commutative case, it is not possible to construct a partial actionangle coordinate chart such that it contains a full neighborhood of a leaf of F. Unfortunately, as Fasso (1995) has pointed out, perturbations to Ho create 'fast' motions (motions of the same order as the perturbation) along the leaves of :F. (A detailed analysis of these motions, in the special case of the Euler-Poinsot rigid body, is carried out in Benettin and Fasso (1996) .) These motions take trajectories out of a locally defined partial action-angle coordinate chart in a relatively short time. Fasso overcomes this problem by showing how to make intrinsic sense of normal forms for the perturbed Hamiltonian. Such normal forms are used to deduce Nekhoroshev estimates (Nekhoroshev, 1977) on the perturbed motions, although to compute and express such estimates a particular atlas of partial action-angle coordinate charts must be fixed . The extent to which these estimates depend on the choice is an open problem (see Fasso (1995, Appendix C)). One alternative approach to non-commutative integrability, due to Mishchenko and Fomenko (1978) (see also Arnold et al. (1988)), is to convert non-commutative integrability into commutative integrability. This is accomplished, at least locally, by 'gluing' together the k-tori to form n-tori; out of these n-tori one builds conventional action-angle coordinates. There is no escaping the difficulties mentioned above, however, because the unperturbed Hamiltonian will still depend on only k < n actions, leading to 'fast' motions in the perturbed system that take trajectories out of locally defined coordinate charts as before. Moreover, this degeneracy of the Hamiltonian appears mysteriously because its GENERALIZED NEKHOROSHEV ESTIMATES 3 source, namely the non-commutative geometry of the original problem, is concealed in the 'commutative' coordinates employed. Integrability through symmetry Non-commutative integrability arises frequently in applications and is often manifest in the form of a non-Abelian symmetry group G. In that case the leaves of F correspond to orbits of G in the phase space 5 . Generally the geometry of the symmetry is obscured in partial action-angle coordinate charts since, as we remarked above, such charts need not contain full neighborhoods of leaves of F 6 . An alternative that we explore here is to substitute for a canonical model space a Hamiltonian G-space normal form. One can view these normal forms as 'non-canonical coordinates' built directly out of the non-Abelian group action. The simplest of these normal forms is a generalization of action-angle coordinates that we shall refer to as action-group coordinates. The 'group' in action-group coordinates can be any compact connected Lie group G; when G = 1'n, one recovers conventional actionangle coordinates. In action-group coordinates the k-tori discussed above are represented by cosets in G of some maximal torus T C G. Action-group coordinates appear in Dazord and Delzant (1987, Section 5). Forerunners of these coordinates have been obtained by Marsden (1981), Gotay (1982), Marie (1983a) and Guillemin and Sternberg (1984, §41). We shall see that in the perturbation analysis of a non-Abelian symmetry, a single action-group coordinate chart suffices, and applies under conditions that are readily verified . Generalized N ekhoroshev estimates The main objective of this dissertation is to demonstrate that a Hamiltonian G-space normal form can indeed be used as a geometric framework for perturbation theory. We 5 6 A proof of this fact will be recalled in Remark 3.12. ln fact (see Remark 4.4), if G acts freely and is compact connected and non-Abelian, then it is n ever possible for a partial action-angle coordinate chart to contain a neighborhood of a G-orbit (i .e., leaf of :F) . GENERALIZED NEKHOROSHEV ESTIMATES FIGURE 4 3. Schematic representation of a solution curve t t----+ Xt of the per- turbed Hamiltonian H, projected onto 'momentum space' fl* using the momentum map J : P -+ g*. The co-adjoint orbit through the initial point J (xo) (denoted 0) is depicted as a sphere. do this in Part 1 by generalizing a well-known theorem of Nekhoroshev (1977) to a 'noncanonical' setting that includes action-group coordinates (Theorem 5.9). As a corollary we deduce a Nekhoroshev-type estimate on the evolution of momentum maps, in a class of integrable Hamiltonian systems with nearly G-invariant Hamiltonians (Corollary 7.1). This result may be informally described as follows . Suppose that a system with Hamiltonian Ho possesses a symmetry group G. Then, under appropriate hypotheses, the system will possess a corresponding conservation law. This law is embodied in the existence of a vector-valued function J : P -+ fl* (g denoting the Lie algebra of G) known as a momentum map that is constant on solution curves t t--+ Xt Next, consider a perturbation to Ho of the form H=Ho+EF, ON THE CONSTRUCTION OF ACTION-GROUP COORDINATES 5 where F is arbitrary. Furthermore, assume that the symmetry is sufficiently large so as to enforce integrability in the unperturbed system 7 . Let t be a solution of the f-----+ Xt perturbed Hamiltonian H , and let O C g* denote the co-adjoint orbit through the initial point J ( x ) 0 E g*. Then we show that there exist positive constants a, b, c, t and r 0 , 0 such that for all sufficiently small E ?: 0, one has (see Fig. 3) Here IJ ( Xt) - OI denotes the distance of J ( xt) from the orbit O, measured using some Ad* invariant inner product on g*. The estimate holds provided the Hamiltonian is real-analytic and satisfies an appropriate 'convexity' condition. One must also assume that action-group coordinates can be constructed in an appropriate neighborhood of x . 0 Sufficient conditions will be formulated precisely (see below). The constants appearing in the estimate depend on the unperturbed Hamiltonian Ho, on the magnitude and analyticity properties of F, and on characteristics of the symmetry group G. In other words, under appropriate conditions the perturbed dynamics, when projected onto 'momentum space' g* by J, evolves exponentially slowly in those directions transverse to the co-adjoint orbits. It turns out that in directions tangential to the orbits, relatively 'fast' motions (order E) are possible. These fast motions correspond to those observed by Fasso described above. On the construction of action-group coordinates The local existence of action-group coordinates has been proven by Dazord and Delzant (1987, Section 5) for suitable non-commutatively integrable systems. We briefly survey this and related work at the end of Chap. 3. In Part 2 we will offer an alternative construction that we have applied to several mechanical systems. In this approach the construction of action-group coordinates is reduced to the construction of conventional action-angle coordinates in an associated lower dimensional phase space known as a symplectic cross 7 To be precise, we require that the Marsden-Weinstein reduced spaces be zero-dimensional; see Chap. 3. In some cases the two-dimensional case can be treated also ; see Remark 3.9. ON THE CONSTRUCTION OF ACTION-GROUP COORDINATES 6 section. This approach may be useful to practitioners of perturbation theory, who are al- ready intimately familiar with conventional action-angle coordinates. The construction may be regarded as a particular application of the so-called 'sym plectic cross section theorem' (Guillemin, Lerman and Sternberg, 1996; Guillemin and Sternberg, 1984). Action-group coordinates can be constructed globally using the cross section method precisely when action-angle coordinates can be constructed globally in the symplectic cross section. Necessary and sufficient conditions for the existence of global action-angle coordinates in a Hamiltonian system are already known (Duistermaat, 1980). In Part 2 we will apply the symplectic cross section technique to the axisymmetric Euler-Poinsot rigid body. Perturbations to this problem have been studied already by Benettin and Fasso (1996) using partial action-angle coordinate charts. Unfortunately to construct action-group coordinates in the neighborhood of a point, one must assume that its image under the momentum map is a regular point of the coadjoint action. In addition, the symmetry group must be acting freely. Nevertheless there do exist nontrivial examples for which action-group coordinates can be constructed, as the axisymmetric Euler-Poinsot rigid body demonstrates. Other examples include the problem of geodesics on 5 2 (and hence the 'regularized' 2D Kepler problem 8 ), the 1 : 1 resonance, and the problem of hydrodynamics of three point vortices on 5 2 . Moreover, it seems likely that techniques such as those outlined here (both the analytic and geometric) will generalize to cases where more sophisticated Hamiltonian G-space normal forms are applicable. We discuss this possibility further in Chap. 13. 8 See Moser (1970) . Part 1 Dynamics Outline of Part 1 In this first Part we focus on dynamics. Underlying geometric constructions will be considered in more detail in Part 2. After dispensing in Chap. 1 with some Lie theoretic preliminaries, we begin in Chap. 2 with a resume of the action-group coordinate framework. We state in Chap. 3 conditions ensuring the existence of action-group coordinates in a given system. We include a description of geodesic motions on S 2 , where the existence of actiongroup coordinates is fairly transparent . Chap. 4 describes the geometry of the unperturbed dynamics of a system in action-group coordinates, as well as the dynamics obtained after naively applying 'multi-phase averaging' to the perturbation. This will explain, from the symmetry point of view, the origin of the 'fast' motions tangent to the co-adjoint orbits, and motivate the Nekhoroshev estimates for the transverse motions, which we deduce in Sections 5- 7. By abstracting Lochak's proof of Nekhoroshev's theorem (Lochak, 1992; Lochak, 1993), we are able to relegate most of the notoriously tedious arguments to an appendix (Appendix A). This abstraction (Theorem 5.9) is also of some independent interest, however (see Chap. 13). There are opportunities for further investigation along the lines initiated in this thesis. We discuss some of these in Chap. 13. CONTENTS OF PART 1 Chapter l. Preliminaries 9 Chapter 2. Action-group coordinates 12 Chapter 3. On the existence of action-group coordinates 20 Chapter 4. Naive averaging 30 Chapter 5. An abstract formulation of Nekhoroshev's theorem 38 Chapter 6. Applying the general theorem to action-group coordinates 46 Chapter 7. Nekhoroshev-type estimates for momentum maps 62 Appendices (Parts 1 and 2) 119 8 CHAPTER 1 Preliminaries The purpose of this chapter is to recall some basic Lie theoretic results and to establish some associated terminology and notation. Weyl chambers When one replaces the torus 1rn in the action-angle model space 1rn x ]Rn with a compact connected Lie group G, the natural generalization of the action space ]Rn turns out to be a Weyl chamber of G. We now recall the standard 'geometric' definition of this object. 1.1 Definition If a group G acts on a manifold X, then an orbit in X is regular if there exist no orbits in X of strictly greater dimension. A point x E X is called regular if it lies on a regular orbit. Let fl denote the Lie algebra of a compact connected Lie group G and let t C fl be any maximal Abelian subalgebra. Denote by flreg C fl the regular points of the adjoint action of G, g ·( = Ad 9 ( (( E fl). A connected component to of the set tnflreg is called an (open) Weyl chamber of G in fl. Some related Lie theoretic facts needed in the sequel are summarized below. 1.2 Theorem Let G be a compact connected Lie group. Then: 1. flreg C fl is open and dense. 2. All maximal tori of G are conjugate. 3. Every g E G lies in some maximal torus. 4. The Lie algebra t of any maximal torus TC G is a maximal Abelian subalgebra. 5. Every point ( E fl belongs to at least one maximal Abelian subalgebra. 6. The map sending a maximal torus to its Lie algebra is a bijection between the max- imal tori of G and the maximal Abelian subalgebras of fl. If t is a maximal Abelian 9 WEYL CHAMBERS 10 subalgebra, then tnflreg Ct is open and dense, and the isotropy group Ge is identical for all points t E t n flreg• This group Ge is precisely the inverse oft under the above correspondence, i.e., Ge is the unique maximal torus whose Lie algebraist. 7. Each regular adjoint orbit intersects each Wey} chamber in exactly one point. 8. If tis a maximal Abelian subalgebra and we define t.L = [fl, t], then one has the direct sum decomposition fl=t EB t .l . Proofs of the above facts can be found in, e.g., Brocker and tom Dieck (1985). The reader may view these facts as generalizations to compact groups of familiar properties of the rotation group S0(3). For example, 1.2.4 corresponds to Euler 's theorem that every rotation is a rotation about some fixed axis. Henceforth G denotes a compact connected Lie group with Lie algebra fl. Let (t, rJ) .-+ t · rJ be some Ad-invariant inner product on fl ( recall that such products always exist since G is compact 1). Then Vt, rJ E fl, Vg E G . 1.3 In particular, we have the infinitesimal version of 1.3, 1.4 Vt,rJ,(Efl. The inner product on fl has a unique extension to a (non-degenerate) C-bilinear form on its complexification fie = fl ®JR C, which we also denote by (t, ry) .-+ t · ry. The identities 1.3 and 1.4 generalize: (t, rJ E fie ; g E G) 1.5 (E, rJ, ( E fie) 1 To obtain one, simply average an arbitrary inner product over G using the Haar probability measure. WEYL CHAMBERS IN g* 11 Weyl chambers in g* Fix a maximal torus T and corresponding maximal Abelian subalgebra t. Equation 1.4 says that adc : g ➔ g is skew-symmetric with respect to the Ad-invariant product. Its image and kernel are therefore orthogonal for any ( E t. This fact, and the commutativity of t, easily show that the decomposition 1.2.8 is orthogonal (explaining the notation t..L). If we define!:= Ann t..L (Ann denotes the annihilator) and !:..L = Ann t, then we obtain the dual decomposition 1.6 g* =!: EB !:.1 . Let <p : g ➔ g* denote the isomorphism induced by the Ad-invariant inner product. Then the orthogonality of the decomposition 1.2.8 implies <p(t) = !: 1.7 The isomorphism <p is G-equivariant if we let G act on g* via the co-adjoint action, g • µ = Ad;-1 µ. It thus establishes an equivalence between the adjoint and co-adjoint representations. In particular, by virtue of l. 7, <p maps a Weyl chamber to C t to a connected component of!: n g;eg, which we call a Weyl chamber in g*. Here g;eg denotes the regular points of the co-adjoint action. The equivalence of the two representations establishes analogues of 1.2.5, 1.2.6 and 1.2. 7 for the co-adjoint action: 1.8 Corollary l. For each µ E g* there exists a maximal Abelian subalgebra t C g such that µ E !: = Ann[g, t]. 2. Let W be a Weyl chamber in g*, i.e., a connected component of!: n g;eg, for some maximal Abelian subalgebra t C g. Then Gµ, = T for all µ E W, where T is the maximal torus with Lie algebra t. 3. Each regular co-adjoint orbit intersects each Wey} chambering* in exactly one point. CHAPTER 2 Action-group coordinates In this section we describe the action-group model space. Chap. 3 will state conditions under which this model space is realizable in a particular system, and give a brief historical sketch of its origins. Let G denote a (real-analytic) compact connected Lie group and let ere denote its complexification (see, e.g., Brocker and tom Dieck (1985)). For computing estimates later on we need to assume that G is realized as a real-analytic subgroup of SO(nc, JR) for some integer nc. Since G is compact and connected this is always possible, by a corollary of the Peter-Weyl theorem (see, e.g., op. cit, Theorem 4.1 and Exercise 4.7.1). We may then identify the Lie algebra of G with a subalgebra fl C JRnoxno. The adjoint action can be written as Ad 9 ~ = g~g- 1 , and the Lie bracket as [6,6] = ad~ 1 6 = 66 - 66- The complexification cc of G can be identified with a complex subgroup of SO(nc, C). The Lie algebra fire of ere is identifiable with the complex subalgebra fl EB if! C cn° xno. 2.1 Remark Although a group like G = SU(2) is a real Lie group, it is ordinarily re- alized as a subgroup of e2 x2 . According to the preceding assumptions, we first need to realize this group as a subgroup of JR 4 X4 ( using the standard identification C2 ~ JR 4 ). The complexification cc will then be realized as a complex subgroup of e4x 4 . Henceforth T C G denotes a fixed maximal torus, t its Lie algebra, and W C i a Wey] chamber in fl*. The model space and its symplectic structure The natural projection fl * --+ t* (the dual map of inclusion) has kernel i..l and thus restricts, by virtue of 1.6, to an isomorphism i : i --+ t*. This map identifies W with an open set to= i(W) Ct*. One calls to a Wey! Chamber also. 12 THE MODEL SPACE AND ITS SYMPLECTIC STRUCTURE 13 We define the action-group model space for a compact connected Lie group C as C x ~. A natural symplectic structure w0 on C x t;, generalizing the canonical structure I : j dqj I\ dpj on '][' n x !Rn, is given by the following proposition : 2.2 Proposition Equip T*C with its natural symplectic structure. For each a E 9*, let aa denote the left-invariant one-form on C with aa(ida) = a (here viewing one-forms as sections of the cotangent bundle) . Then the embedding C x ~ c.....+ T*C that maps (g, p) to (i- 1 (p))a(g), maps C x ~ onto a symplectic submanifold of T*C. We define w0 to be the symplectic structure on C x ~ pulled back by this embedding. Note that w0 does not depend on the choice of Ad-invariant inner product on fl , or on the realization of C as a linear group . The proof of 2.2 will follow from a more general observation we shall make in Part 2. The injection <p- 1 oi- 1 : t* c.....+ g maps t* isomorphically onto t (see 1.7), and so identifies ~ with a Wey! chamber to C tin g. For computing estimates later on, it is convenient for us to make the corresponding identification of C x ~ with C x to. We now describe explicitly the symplectic structure on C x to induced by this identification. This symplectic structure of course does depend on the choice of Ad-invariant inner product. Since we need a complexification of the model space to make estimates later on, we develop notation which also makes sense on the complex manifold cc x tC containing C x to. (By tC we mean the complex subspace t EB it of fie= fl EB ig .) For fo E fie, To E tC and (g,p) E cc x tC, define the (complex) vector (fo,70 \ ,P tangent to cc x tC at (g , p) by 2.3 d (fo, To) 9,p = dt (g exp(tfo), p + tTo) . jt=O Note that every tangent vector in T(g, p) (Cc x tC) is of this form. On cc x tC define the one-form 0a by 2.4 and define the two-form wa on cc x tC by 2.5 wa = -d0a HAMILTONIAN VECTOR FIELDS IN ACTION-GROUP COORDINATES 14 We claim: Restricted to G x to C Ge x tC, wa agrees with the (real) symplectic structure w 0 on G X to defined above, after making the identification G x to ~ G x to discussed above. Our claim follows from formulas for w0 (analogous to 2.4 and 2.5) we shall derive in Part 2. These formulas also appear in Dazord and Delzant (1987, Section 5) (who obtain it via a different route). For our applications to perturbation theory, we also need explicit equations of motion and an explicit formula for the Poisson bracket. We turn to these next. Hamiltonian vector fields in action-group coordinates If ~ : Ge x tC -+ ff and T : Ge x tC -+ tC are arbitrary holomorphic maps, we define f vector fields ~ • 9 and T • JP on Ge x tC by (c :g) (g,p) =(~(g,p),0) 9,p (T · :p) (g,p) =(0,T(g,p))g,p f An arbitrary vector field on Ge x tC is then of the form ~ · 9 + T • JP for some vector-valued functions ~ : QC X tC -+ gC and T : QC X tC -+ tC. Write tl.C = tl. EB itl. and define ~ to be the connected component containing to of the set {p E tC I adp : tl.C -+ tl.C is invertible } . For p E ~, let Ap : tl.C -+ tl.C denote the inverse of adp : tl.C -+ tl.C. In particular, let us record that 2.6 and 2.7 EQUATIONS OF MOTION IN ACTION-GROUP COORDINATES 15 It is possible to show that the restriction of the (complex) form wa to cc x t~ is non-degenerate. If H : cc x ~ ➔ C is holomorphic, then the corresponding (complex) Hamiltonian vector field XH is defined by XH J wa = dH. Indeed 2.8 where 2.9 and 2.10 Here CY : fl ➔ t denotes the projection along t1-, and CY1- : fl ➔ t1- that along t. These formulas are the complexified version of formulas on C x to ~ C x ~ whose derivation will be given in Part 2. The vector valued functions ~~ : cc x ~ ➔ fie and ~~ : cc x ~ ➔ tC appearing in 2.9 and 2.10 are defined implicitly by and Here a dot denotes the non-degenerate C-bilinear form on fie. Equations of motion in action-group coordinates Recalling that we are identifying C with a linear group, we may think of elements of C as matrices and use formulas 2.8-2.10 to write the 'equations of motion' corresponding to a Hamiltonian H as 2.11 EQUATIONS OF MOTION IN ACTION-GROUP COORDINATES 16 Although one can (abstractly) make sense of these equations without the realization of Gas a linear group, our estimates later on will depend on the vector space structure of cnG xnG in which we have assumed cc to be embedded. 2.12 Remark It is worth comparing the above equations to those corresponding to Hamil- tonian vector fields on T*G ~ G x fl* (left trivialization, say). Such equations were already known to Cushman (1977) (see Abraham and Marsden (1978, Proposition 4.4.1)). In the present notation, and identifying fl* with fl as above, these equations take the form ( (g, µ) E G x fl ) . Can one use the 'coordinates' G x fl as a geometric framework for studying perturbations to integrable Hamiltonian systems whose phase space is T*G (such as the Euler-Poinsot rigid body)? Our (unpublished) investigations suggest that such an approach fails, as far as Nekhoroshev estimates are concerned, unless the Hamiltonian is both left and right invariant 1 . This is not the case in many interesting examples. To help convince the reader that 2.11 indeed generalizes Hamilton's canonical equations, we point out a simple example. 2.13 Example Take G = S 1 . Then we realize Gas S0(2), i.e., the set of matrices of the form g = [cos0 sin 0 - sin 0] , cos0 It is clear that t = fl C JR 2 x 2 consists of matrices of the form IE JR 1 i.e., invariant with respect to the cotangent lift of pre- as well as post-multiplication in the group . EQUATIONS OF MOTION IN ACTION-GROUP COORDINATES 17 and that the only choice for to is to = t. Also, t.L = 0. Choose for an inner product on f1 the one given by 0 [ Ii - Ii O l · [0 - ]zl = Ii Iz . lz 0 Write H'(0,I) = H(g,p) = H cos . 0 - sin 0] , [O ( [ Slll 0 cos 0 I - I] ) . 0 Then, after making the change of coordinates from (g,p) to (0, I), one sees that 2.11 (in uncomplexified form) becomes 8H' [)J · 8H' I= - 80 ' which are Hamilton's canonical equations on 8 1 x R. One generalizes the above argument to G = 1'n ~ SO(2) x • • · x SO(2) by realizing G as the set of block diagonal 2n x 2n matrices whose 2 x 2 blocks are of the form just described for n = l. One then recovers the familiar form of Hamilton's equations on 1'n X Rn. The next example is relevant to the problem of geodesic motions on 8 2 (see 3.4). 2.14 Example Take G = SO(3). Then f1 = .so(3) can be identified with R 3 via the iso- morphism ( i----+ l : R 3 --t .so(3) defined by lu = ( x u (u E R 3 ). Note that this isomorphism extends uniquely to a C-linear isomorphism C3 --t .so(3, C). Let {e 1, e2, e3 } denote the standard basis of R 3 . Choose T C SO(3) to be the rotations about the e3 axis, so that t = span{e 3} ~ JR. Then t.L = span{e1, e2} ~ R 2. The adjoint action is given simply by g • ( = g( (g E SO(3), ( E R 3), so that the regular adjoint orbits are the spheres centered at the origin of positive radius. Whence f1reg = R 3\ { 0}. The standard inner product a· b = a 1b1 + a2b2 + a3b3 is Ad-invariant. A connected component to oft n f1reg is given by to= {te3 It> O} ~ (O,oo) . So G X to~ SO(3) x (O,oo). THE POISSON BRACKET IN ACTION-GROUP COORDINATES 18 One has le ~ C3 with ad( : flic ➔ flic given by ad( rJ = ( x ry. Fixing p E tlC ~ <C, one finds that adp: t.LIC ➔ t.LIC (t.LIC ~ <C2 ) is given by adp(6,6) = (-p6,P6). Therefore, ~ ~ <C\ {0} and >..P : t.LIC ➔ t.LIC is given by >..p(( 1 , 6) = (6/p, -6/p) (p E <C\ {0} ). The projections a : fl ic ➔ tlC and a.L : fllC ➔ t.LIC are given by a(6, 6, 6) = 6, a.L (6, 6, 6) = The equations of motion 2.11 can now be written as 18H 0 P892 iJ=g 1 8H -p 891 8H Bi 8H p=-8g3 =g 8H Bi 18H P891 8H 18H 0 -p 892 18H 0 -Bi -p 891 P892 18H (g E SO(3,<C)) (p E <C\{O}) where The Poisson bracket in action-group coordinates Our convention for defining Poisson brackets is { u, v} = Xv J Xu J w. If t f--)- Xt is an integral curve of a Hamiltonian vector field XH, then according to this definition for any function u. 2.15 Lemma The Passion bracket on cic x t~ is given by THE POISSON BRACKET IN ACTION-GROUP COORDINATES 19 PROOF. { u, v }(g, p) =< du, Xv(g, p) > au {Ju = ~v(g, P) • {Jg (g, p) + Tv(g,p) • {)p (g, p) by 2.8 8v 1. 8v au 8v au = ({)p(g,p)+ApCT 8g(g,p)) · 8g(g,p)-CT8g(g,p)· 8p(g,p) by 2.9 and 2.10, au 8v 8v au = 8g(g,p). 8p(g,p)- 8g(g,p). 8p(g,p) 1.8v au +ApCT {)g(g,p)· {)g(g,p) The l ast term can be written □ CHAPTER 3 On the existence of action-group coordinates In this section we formulate the main theorem on the existence of action-group coordinates for a Hamiltonian system with a non-Abelian symmetry. A proof of the theorem is postponed to Part 2, where the construction of the coordinates is reduced to the construction of conventional action-angle coordinates in an associated lower-dimensional phase space. A Hamiltonian system is said to possess a (continuous) symmetry when there exists a Lie group G acting on the system's phase space (P,w), with respect to which the Hamiltonian function H0 is invariant: Ho(g • x) = H0 (x) for all g E G and x E P. The group G is said to be acting in a Hamiltonian fashion if it acts by symplectic diffeomorphisms, and if the infinitesimal generators ~P (~ E g) of the action are (global) Hamiltonian vector fields on P. In that case there exists a map J : P -+ g*, called a momentum map, with the property that it delivers Hamiltonian functions Je : P -+ Iii for the generators according to the formula Je( x) = (J (x), 0- Here ( ·, ·) denotes the natural pairing between g* and g. Thus ~P = X1{, where X f denotes the Hamiltonian vector field corresponding to a function f, i.e., the vector field defined through X f J w = df. Noether's theorem states that the functions Je (which Poisson-commute only in the Abelian case) are integrals of motion for the G-invariant Hamiltonian H0 . In particular, J( xt) E g* is constant for all solution curves t M Xt. See Marsden and Ratiu (1994), Abraham and Marsden (1978 , Chapter 4) or Guillemin and Sternberg (1984) for background on momentum maps and for an introduction to the geometric point of view of symmetry in mechanics. For a rapid introduction and recent survey, see Marsden (1992). Unless otherwise indicated notation follows Abraham and Marsden (1978). 20 ORBIT TYPE CONDITIONS 21 System symmetry breaking Let (F, w) be a symplectic manifold and let G be a compact connected Lie group acting on Pin a Hamiltonian fashion, with momentum map J : P-+ g*. We suppose that J is G-equivariant (with G acting on g* by the co-adjoint action). The quadruple (P,w, G, J) (or simply P) will then be called a Hamiltonian G-space. 3.1 Example (Action-angle coordinates) Take P = 1rn x JR.n and w = z'.,dqj I\ dpj, Let G = 1rn = JR. n /21rzn act on P according to 0 • (q, p) = (0 + q, p). This action is Hamiltonian with equivariant momentum map J: F-+ g* ~ JR_n given by J(q,p) = p. A Hamiltonian is G-invariant precisely when it depends only on the action coordinates p. A basic problem of perturbation theory is to understand the dynamics associated with Hamiltonians H : P-+ JR. of the form H= Ho+F, where Ho is G-invariant and F is small in an appropriate sense. Viewing G as a symmetry group for H 0 , one sometimes refers to this as a Hamiltonian system-symmetry breaking problem. Orbit type conditions If a group G acts on a space M, then we denote the point stabilizer (isotropy) subgroup at x E M by Gx, Recall that the orbit type of x is the conjagacy class of subgroups of G with representative Gx, It is denoted (Gx)Reconsider the case of a Hamiltonian G-space as above. In addition to the orbit type of a point x E P we shall also, in a slight abuse of terminology, refer to its co-adjoint orbit type. This is defined to be the orbit type of J (x) E g*. It follows from a general fact about group actions (see, e.g., Bredon (1972)) that if P is connected then there exists an open dense subset of P whose points are of uniform orbit type (called the maximal orbit type). From the point of view of understanding the fate of generic initial conditions in the symmetry breaking problem above, it is reasonable to ORBIT TYPE CONDITIONS 22 restrict attention to this open G-invariant subspace. In all applications of which this author is aware, one can also find an open dense subset of P with constant co-adjoint orbit type. (Whether this holds in general is not immediately clear 1.) Such a subset is also G-invariant, by momentum map equivariance. The preceding arguments suggest that, from the point of view of perturbation theory, an interesting class of Hamiltonian G-spaces is those spaces with simultaneously a constant orbit and co-adjoint orbit type. The simplest case is a space in which G acts freely and J(P) C g;eg· In this case all points in P have orbit type (ida) and (by 1.8.1, 1.8.2 and 1.2.2) co-adjoint orbit type (T), where Tis any maximal torus of G. These are the spaces we shall consider here and are precisely those spaces for which action-group coordinates can be constructed, under an appropriate integrability hypothesis. Note that if G is Abelian, then g;eg = g*, so that the condition J (P) C g;eg is always satisfied in that case. 3.2 Example (Action-group coordinates) Take P =G x to and = (notation as w we in Chap. 2). The group G acts freely on P according tog· (h,p) = (gh,p). We claim that this action admits an equivariant momentum map. Writing <"P 9 (h,p) = g • (h,p), one can deduce from 2.4 that 0a is invariant under pullback by <P 9 , for all g E G. As a consequence the Lie derivative of 0a along any infinitesimal generator ~P (~ E g) vanishes. Therefore, by Cartan's 'magic formula' and 2.5, one has frJ w = d(frJ 8a) Thus the generator ~P is the Hamiltonian vector field corresponding to the function Jf = ~P J 0a. The definition of infinitesimal generators leads, in the notation of 2.3, to the explicit formula 1 To settle the question seems to require a detailed understanding of the structure of the momentum map image. So far only the case of compact P seems sufficiently well-understood; see Kirwan (1984) for a treatment of the compact case and see Hilgert, Neeb and Plank (1994) for partial results in the non-compact case. ORBIT TYPE CONDITIONS 23 One now computes Whence a momentum map JG : P ➔ g* is given by 3.3 Recall that <p maps the Weyl chamber <p- 1 ( p ) E W C in g;eg for all p W in g* bijectively onto the chamber to, so that E to. In particular, JG ( P ) C g;eg, so that ( G x to, w e , G , JG) is a Hamiltonian G-space satisfying our orbit type conditions. Of course Example 3.1 is just Example 3.2 in the special case G = 1rn. 3.4 Example (Geodesic motions on 5 2 ) Consider a point mass M constrained to move on the surface of a smooth sphere of radius L. One identifies the position of M with a point on the unit sphere 5 2 , so that the phase space is T* 5 2 , equipped with its standard symplectic structure. We identify this space with T5 2 ~ {(q, v) E R 3 x R 3 I llqll = 1, q • v = 0} , using the standard metric on 5 2 , in which case the symplectic structure is w = -d0, where Here q0 = d/dtlt=Oqt is to be viewed as an element of R 3 . The Hamiltonian is H 0 (q, v) = L 2 llvll 2 /(2M), which is G-invariant with respect to the action of G = SO(3) defined by g • (q, v) = (gq, gv). This action is free and has an equivariant momentum map J : P --+ g* ~ R 3 given by J(q, v) = q xv. We restrict attention to the open dense G-invariant subset of phase space obtained by removing the zero section: p = { (q' V) E T 5 2 I y -/ 0} Then J(P) = R 3 \{0} = g;eg, so that (P,w,G , J) is a Hamiltonian G-space satisfying our orbit type conditions. INTEGRABILITY 24 3.5 Remark Bearing in mind Poincare's model of S0(3), S0(3) ~ {(q, v) E T5 2 I llvll = 1} , one observes in Example 3.4, that P can be identified with S0(3) x (0, oo), the action-group model space for G = S0(3) (see 2.14). In fact this identification is symplectic2 . Whence it gives a (rather straightforward) realization of action-group coordinates for the problem. Incidentally, in these coordinates the Hamiltonian is given by H 0 (g, p) = L 2 p 2 /(2M), (g,p) E S0(3) X (0, oo). Also note that neither conventional action-angle coordinates, nor partial action-angle coordinates, can be constructed globally on P in this problem. Another simple example satisfying our orbit type conditions is the 1 : 1 resonance, where P = R 4 \ {0} and G = SU (2). More complicated examples include the problem in hydrodynamics of three point vortices on 5 2 (G = S0(3) x 5 1), and the axisymmetric Euler-Poinsot rigid body to be discussed in Part 2 (G = S0(3) x 5 1). In both cases one needs to restrict to some open subset of the phase space to ensure the above orbit type conditions. In more complicated examples one or both of the above conditions on the orbit types often fail. For example, in the problem of geodesic motions on 5 3 (see Chap. 13) the symmetry group S0(4) fails to act freely (although most points do have co-adjoint orbit type (T), T C SO (4) denoting the maximal torus). In the 1: 1: 1 resonance (P = R 6 = c3, G = SU(3); see Part 2), all non-zero points have orbit type (SU(2)) and co-adjoint orbit (SU (2)), so that both orbit type conditions fail. These comments not withstanding, the above orbit type conditions are the natural ones to study first. Integrability In the case that G acts freely, a reduction in dimension of a Hamiltonian system with symmetry is achieved by Marsden- Weinstein reduction: 2 For a proof, see Part 2. INTEGRABILITY 25 3.6 Theorem (Marsden and Weinstein (1974), Meyer (1973)) Let (P, w, G, J) be a Hamiltonian G-space and assume G acts freely and properly. Then for anyµ E J(P), J- 1 (µ) is a Gµ-invariant submanifold of P and Pµ = J- 1 (µ)/Gµ admits the structure of a smooth manifold, with respect to which the natural projection 1 µ: J- 1 (µ)-+ Pµ is a surjective submersion. There is a unique symplectic form wµ on Pµ with the property that ,;wµ = i:w, where iµ : J- 1 (µ) -+ P is the inclusion. The symplectic manifolds Pµ (µ E J(P)) are called the reduced spaces. Of course, as we assume G is compact, all actions of G are proper. If Ho is a G-invariant Hamiltonian, then for each µ in Theorem 3.6, J- 1 (µ) C P is an invariant submanifold (by Noether), and there exists a Hamiltonian Hµ: Pµ-+ JR such that the vector fields XH0 1J- 1 (µ) and XHµ (defined by XHµ J wµ = dHµ) are ,µ-related . It can be shown that if the integral curves in the reduced space Pµ of XHµ are known, then the integral curves of XHo lying in J- 1 (µ) C P can be reconstructed by solving linear ordinary differential equations with time dependent coefficients (Marsden, Montgomery and Ratiu, 1990). If J(P) C g;eg then the reduced spaces all have dimension d = dimP- (dimG + rankG) . Here rank G denotes the dimension of the maximal tori of G. If d = 0 or d = 2 then Hamilton's equations on the reduced spaces can always be 'solved': In the d = 0 case integral curves of XHµ are trivial (and the reconstruction equations are in fact autonomous); in the d = 2 case integral curves of XHµ coincide with level sets of the reduced Hamiltonian Hµ. It is natural to call the original system integrable in either case: 3.7 Definition A Hamiltonian G-space (P,w, G, J) , with G acting freely and J(P) C fl;eg, will be called G-integrable (or geometrically integrable) if d = 0, and (G, H 0 )-integrable (or dynamically integrable) if d = 2. In either case we say that the integrability is noncommutative if G is non-Abelian. There are, of course, more general notions of non-commutative integrability (see, e.g., Dazord and Delzant (1987)). In the present context the above definition is appropriate. THE EXISTENCE THEOREM 26 In this paper we will be restricting attention to geometric integrability. 3.8 Examples The problem of geodesic motions on S 2 (Example 3.4) is geometrically in- tegrable. Action-group coordinates (Example 3.2) also constitute a geometrically integrable space. 3.9 Remark Restricting attention to the geometrically integrable case does not entirely rule out treatment of dynamically integrable systems. Under certain conditions one can enlarge the action of G, in a (G, H 0 )-integrable space, to an action of G x S1, in such a way that the space is G x S 1 -integrable. The basic idea is to 'compactify' the action of JR generated by XHo. In that case the G-invariance of the Hamiltonian Ho extends to G x S 1 -invariance, by energy conservation. For example, the Euler-Poinsot rigid body is dynamically integrable with respect to an S0(3) symmetry. If one avoids certain hyperbolic invariant manifolds, then the preceding compactifying procedure delivers SO (3) x S 1-integrability. In the special axisymmetric case the extra S 1 -action corresponds to the symmetry of the body. See Part 2 for details. The existence theorem The following result states conditions under which action-group coordinates can be constructed in a Hamiltonian G-space. Recall that <p : g -+ g* denotes the isomorphism corresponding to the fixed Ad-invariant inner product on g. 3.10 Theorem Let G be a compact connected Lie group, and let (P, w, G, J) be a Hamil- tonian G-space on which G is acting freely, and far which J(P) C g;eg . Assume the space is G-integrable in the sense of 3. 7. Assume that P is connected, and that each fiber of J : P -+ J(P) is compact or has a finite number of connected components. Assume that U = <p- 1 (J (P)) n to (which is open in to) is smoothly contractible. Let G act on G x U C G x to as in Example 3.2. Then there exists an equivariant symplectic diffeomorphism ¢ : G x U -+ P such that Jo ¢ = JG, where JG : G x U -+ g* is the momentum map defined by 3.3. HISTORICAL REMARKS 27 This theorem holds in either the C 00 or real-analytic categories. A proof will appear in Part 2. 3.11 Remark The contractability hypothesis can be weakened if the fibres of J are known to be connected (see Part 2). On the other hand, if U in 3.10 fails to be contractible, then one may always work 'locally' by replacing P by a connected component of J- 1 (G • V), for some appropriate contractible set V C U. This suffices for the applications to perturbation theory in the sequel. 3.12 Remark Let (P, w, G, J) be a Hamiltonian G-space on which G is acting freely and for which J(P) C g;eg· In that case Gµ is Abelian (a maximal torus in fact) for allµ E J(P). In Part 2 we shall recall the following formula (following from the equivariance of the momentum map): w(fr(x), ryp(x)) = (J(x), [~, 77]) (xEP,~,77Eg). Since G µ is Abelian (µ E J (P)), the restriction of w to Gµ-orbits vanishes. It follows that the distribution Don P defined by D(x) = (Tx(GJ(x) • x))w is coisotropic (see the Introduction for a definition). Assume Pis G-integrable. Then dimJ- 1 (µ)/Gµ = 0 for anyµ E J(P), so that D = (kerTJ)w. But from the definition of a momentum map, g* ?- P !.+ P/G has the 'dual pair' property (p denoting the natural projection), i.e. kerTJ and kerTp are w-orthogonal distributions. In particular, D = kerTp, i.e., Dis the distribution tangent to the foliation :F of P by G-orbits. This foliation, as we have just demonstrated, is coisotropic. Furthermore, as Dw = ker T J, :Fis also symplectically complete. AG-invariant Hamiltonian Ho : P ➔ ~ is therefore integrable in the sense of the Introduction. Historical remarks Recall (see the Introduction) that a rather general notion of integrability, geometrically formulated, is that of a symplectically complete coisotropic foliation :F. This foliation is to have the property that the Hamiltonian of interest is constant on its leaves. This formulation HISTORICAL REMARKS 28 of integrability has its origins in the work of Alan Weinstein and Paulette Libermann; see Dazord and Delzant (1987) for details. Under the hypotheses of Theorem 3.10, the orbits of G form the leaves of a symplectically complete coisotropic foliation (see Remark 3.12 above). There are two complementary approaches to studying the geometric structure of a foliation F as above. One may focus either on neighborhoods of the (coisotropic) leaves of F, or on neighborhoods of leaves of the symplectic orthogonal foliation Fw (which is isotropic3 ). The latter problem, as we have described in the Introduction, was solved by Nekhoroshev (1972) (under the hypothesis of compact leaves), and clarified later in Dazord and Delzant (1987). Attempts to address the former problem, which are more recent, begin with independent studies of Gotay (1982) and Marie (1982; 19836), which classify the neighborhood of an arbitrary isolated coisotropic submanifold. The complementary problem of classifying the neighborhood of an isotropic submanifold, and the 'overlapping' problem of classifying the neighborhood of a Lagrangian 4 submanifold, were already solved in Weinstein (1981) and Weinstein (1971) respectively. In the studies of Gotay and Marie, it is shown that the neighborhood of a coisotropic submanifold M C P looks like a neighborhood of the zero section of the dual E* of its characteristic bundle E, equipped with an appropriate symplectic structure. The characteristic bundle of M is defined by E ={v E TM I w(v, w) = 0 Vw E TM}. In the special case of symmetry considered m Theorem 3.10, E(x) = Tx(Gµ • x), where µ =J(x). The symplectic diffeomorphism realizing a neighborhood of M in P as a neighborhood of the zero section of E*, and the associated sym plectic structure bestowed on E*, both 3 4 Meaning that the tangent distribution of Fw is contained by its symplectic orthogonal. A submanifold is Lagrangian if it is simultaneously isotropic and coisotropic, or equivalently if spaces tangent to the submanifold coincide with their symplectic orthogonals. HISTORICAL REMARKS 29 depend on a choice of splitting TM= E EB V. However, up to neighborhood equivalences, all choices lead to the same answer. In Marle (1983a) the case where M is the orbit of a free 5 and geometrically integrable Hamiltonian action is studied. Here it is pointed out that the choice of splitting TM = EEB V can be reduced to the choice of a splitting fl = QEB b, where Q denotes the Lie algebra of some subgroup H C G representing the co-adjoint orbit type (H) of points in M. Marle's description is more general than action-group coordinates (which correspond to the case H = T), but lacks the same concreteness in the sense that the bundle E* remains an abstractly defined object. Dazord and Delzant (1987, Sect. 5) show that in the special 'regular' case (H = T), the bundle E* trivializes: E* ~ G x U (UC JRk). Moreover, they show that U is naturally identifiable with an (open) Weyl chamber to, the appropriate symplectic structure on G x to being -d0a, where 0a corresponds, under the identifications discussed in Chap. 2, to the one-form 0a defined in 2.4. The equations of motion 2.11 and the formula of Lemma 2.15 for the Poisson bracket do not appear in Dazord and Delzant (1987), or elsewhere, as far as we are aware. We are also unaware of work, outside that to be presented in Part 2, in which action-group coordinates are constructed in concrete examples. One other approach to constructing 'coordinates' in the neighborhood of the orbits of a G action, is to apply the technology of symplectic cross sections (Guillemin et al., 1996; Guillemin and Sternberg, 1984) alluded to in the Introduction. In particular, this technique is well suited to addressing global existence questions. An application of this technology in the special case of action-group coordinates is to be expounded in Part 2. 5 The case of non locally free actions appears to be unexplored. CHAPTER 4 Naive averaging Consider a Hamiltonian G-space P satisfying the hypotheses of Theorem 3.10 and let H : P---+ JR be a Hamiltonian of the form H=Ho+F, where Ho is G-invariant and F is some perturbation . Then according to the theorem there exists an open set U C to and an equivariant symplectic diffeomorphism q>: G x U---+ P . The equivariance implies that (Hoo cp)(g,p) = h(p) for some h: U---+ JR. Thus in action-group coordinates a Hamiltonian system-symmetry breaking problem of the above form translates into the problem of studying a Hamiltonian H : G x U ---+ JR of the form H(g,p) = h(p) + F(g,p) . 4.1 The first task of this section is to describe the dynamics associated with the unperturbed part H0 (g , p) = h(p) of such a Hamiltonian. We will then describe the dynamics associated with an averaged form of the perturbed Hamiltonian H. This will highlight the role played by the symmetry in determining the nature of the perturbed dynamics , as well as motivate the Nekhoroshev estimates developed in subsequent sections. The unperturbed dynamics According to 2.11 the equations of motion for a Hamiltonian Ho(g,p) = h(p) are g = g0,(p) p=O , where 0-(p) = '\lh(p) Et. We conclude that all integral curves t t-----+ (gt,Pt) of XH 0 are of the form 9t = goe tno Pt= Po , 30 THE UNPERTURBED DYNAMICS T-orbit 31 G-orbit C;,! G u FIGURE 1. Cartoon of the unperturbed motions in G x U C G x to, depicting the foliation by invariant G-orbits, and the finer foliation by invariant Torbits. where S1 0 = D(po), In particular, each G-orbit G x {po} (Po E U) is an invariant manifold for the flow, and the restriction of the flow to G x {p0} ~ G is of the form g0 t-t g0 etno. This generalizes the situation in classically integrable systems where one has a foliation by invariant tori, each supporting a quasi-periodic motion whose frequency D(p) is delivered by a similar formula. Corresponding to the identification G x {p0} ~ G we have an identification of XHo IG x {p0 } with the left-invariant vector field on G corresponding to S1 0 E t C fl· It follows that this vector field on G corresponding to XHo I G x {po} is tangent to T and every left coset gT. If we let T act on G x U according to q·T (g, p) = (gq, p) (q E T), then these cosets of T in G correspond to T-orbits in the phase space G x U. Thus we have a finer foliation of the phase space G x U by invariant tori (see Fig. 1), but these tori have strictly less than half the dimension of phase space in the non-Abelian case. Let k = dim t. The k-dimensional Z-module 4.2 I = {p E t I exp (p) = id} is called the integral lattice of T in t. We have: THE UNPERTURBED DYNAMICS 32 An integral curve of XH0 lying in G x {p0 } is periodic if and only if Q, 0 E vl = {vn I n E J} for some v > 0, in which case a (not necessarily minimal) period is l/v. 4.3 Definition Identify t with JR.k, by choosing some Z-basis for I, and call an element v oft~ JR.k irrational if I:j Vjnj i- 0 for all non-zero n E zk. We leave it to the reader to check that the choice of basis in this definition is immaterial. We have: The integral curves lying on G x {p0} densely fill each of the T-orbits foliating G x {p0 } if and only if Q 0 E tis irrational. 4.4 Remark The action of T on G x U discussed above is in fact Hamiltonian, with equivariant momentum map jG: G x U------+ t* given by jG(g,p) = (i o cp)(p). For a proof, see Part 2. The fibers of jG are the G-orbits, which constitute the leaves of a coisotropic foliation Fon G x U, by Remark 3.12. Since the space is geometrically integrable, Remark 3.12 also implies that F is symplectically complete and that the leaves of the associated w-orthogonal foliation Fw are connected components of fibers of JG. The fibers of JG are in fact connected (indeed they are precisely the orbits of the T action discussed above; see Lemma 4.5.2 below), so that the maps constitute a concrete realization of the foliations F and Fw as fibers of a dual pair 1 . Since Fis a symplectically complete coisotropic foliation with compact fibers, one may construct (see, e.g., Dazord and Delzant (1987)) an atlas of partial action-angle coordinate charts on G x U. In each chart i A pair of Poisson maps Qi ~ P ..!::!:+ Q2 (Qi, Q2 Poisson manifolds) is a dual pair if the tangent maps Tpi and Tp2 have w-orthogonal kernels. THE AVERAGED DYNAMICS 33 the foliation by T-orbits will be represented by the trivial foliation But if G is a non-Abelian compact connected Lie group, the bundle G ➔ G /T is topologically nontrivial. Since the T-orbits lying in a G-orbit G x {p} ~ G (p E U) correspond to left cosets of T in G, we conclude that no partial action-angle coordinate chart can be constructed in the non-Abelian case such that it contains an entire neighborhood of some G-orbit. The averaged dynamics Suppose that a Hamiltonian system with Hamiltonian Ho admits a foliation F by closed (compact and boundary less) invariant manifolds, and assume that the restricted flow on a generic leaf is ergodic 2 . The familiar example is the foliation by invariant tori of an integrable system in action-angle coordinates, provided the Hamiltonian Ho (q, p) = h(p) is nondegenerate. Furthermore, assume (as in this example) that each leaf of :F supports a probability measure invariant with respect to the restricted Hamiltonian flow, and varying from leaf to leaf in some smooth fashion. Given a perturbed Hamiltonian H, one can form a 'first order approximation' fl to H, constructed by averaging H over the leaves of F using the measures. One then expects the flow of fl to be a reasonable approximation to that of H (at least better than that of Ho). Turning this heuristic expectation into rigorous theorems is, more or less, the principal preoccupation of the classical perturbation theory 3 . The necessity of ergodicity is well-known. We now carry out the averaging procedure just outlined m the case of an integrable Hamiltonian system in action-group coordinates. Note that in the example of action-angle coordinates mentioned above the averaged Hamiltonian fl is itself integrable, and therefore not terribly interesting. 2 By 'generic', we mean with respect to some appropriate Borel measure on the factor space, which we suppose is Hausdorff. 3 See also the introduction to the book by Lochak and Meunier (1988). THE AVERAGED DYNAMICS 34 Assume that the unperturbed Hamiltonian H0 (9, p) = h(p) in 4.1 is nondegenerate, i.e. that the map p f--t f2(p) : U-+ tis a local diffeomorphism . Then for almost any p E U (in the sense of Lebesgue measure), the frequency Q(p) is irrational (in the sense of 4.3). It follows that we may take the leaves of :F to be the orbits of the T-action qr(9,P) = (9q,p) discussed above. We therefore define fl(9,p) =1r H(9q,p)dµ(q) , whereµ is the unique translation-invariant probability measure on the torus T. By construction fl is T-invariant. Our objective is to Poisson-reduce (see, e.g ., Marsden (1992)) the dynamics of fl to dynamics on an appropriate realization of the quotient space (G x U)/T. Recall that the action of G on G x to admits an equivariant momentum map JG : G x to -+ £1* defined by 3.3. 4.5 Lemma l. The fibers of JG are the T-orbits. 2. If O C £1* is a co-adjoint orbit, then (JG)- 1(0) is some G-orbit G x {p} (p E to). PROOF . By 3.3, two points (9 1,p 1), (92,P2) E G x to lie in the same fiber of JG if and only if 4.6 Since p 1 and pz lie in the same Wey! chamber to, it follows from Theorem 1.2.7 that 4.6 is equivalent to Pl= P2 and The requirement 92191 E Gp 1 is equivalent to 91 and 92 belonging to the same left coset of Gp 1 in G. Since P1 E to Ctn f1reg, Theorem 1.2.6 implies that Gp 1 = T. The preceding arguments show that the fibers of JG consist of those sets of the form 9T x {p} (9 E G, p E to), which proves l. By 3.3, JG maps a G-orbit G x {p} onto the co-adjoint orbit through p E to. By the equivariance of JG , the preimage J- 1 ( 0) is a union of G-orbits in G x to. To show J- 1 ( 0) THE AVERAGED DYNAMICS 35 is in fact a single orbit it therefore suffices to show that distinct G-orbits cannot be mapped to the same co-adjoint orbit. But JG maps an orbit G x {pi} and an orbit G x {p 2} to the co-adjoint orbits through P1 and P2 respectively. By Theorem 1.2.7 these orbits are the same if and only if P1 = pz. □ Note that Lemma 4.5.1 says that we have a natural way of identifying the abstract quotient (G x U)/T with V = JG(G x U) C fl*, the momentum map JG : G x U -+ V being a realization of the natural projection G x U-+ (G x U)/T. Being an equivariant momentum map, JG is also a Poisson map, if we equip fl* with the positive Lie-Poisson structure{·, ·}+ (see, e.g., Marsden and Ratiu (1994, Chapter 10)). If u is a smooth function on fl*, then by definition its corresponding Hamiltonian vector field Xu on fl* is the vector field satisfying the equation Xu J df = {f, u}+ for all smooth f : fl* -+ JR. Hamiltonian vector fields on a Poisson manifold are always tangent to the symplectic leaves, which in this case are the co-adjoint orbits. If a function is constant on a leaf, its Hamiltonian vector field vanishes. The functions H, Ho and P on G x U are T-invariant. They therefore drop, via JG : G x U -+ V, to functions H', Hb and F' on V C fl*. Since Ho is G-invariant, 4.5.2 guarantees that Hb is constant on the co-adjoint orbits, and hence that XH~ = 0. Therefore, Xfl, = Xp, . Since JG is a Poisson map, the vector fields X fl and X fl' are JG-related. Therefore, JG maps integral curves of X fl onto integral curves of X p,. If F = 0, then X p , = 0. This is consistent with Noether's theorem, which states that JG(xt) E fl* is constant for all integral curves t M Xt of Xfl, if fl is G-invariant. If F =/=- 0, then X p, need not be trivial, or even integrable, but must nevertheless remain tangent to the co-adjoint orbits. It follows from 4.5.2 that the G-orbits G X {p} (p E U) persist as invariant manifolds of X fl· In other words, the action variables p E U are integrals of motion for the averaged Hamiltonian fl. THE AVERAGED DYNAMICS 36 T-orbit integral curve of X fl G-orbit integral curve of X p, r co-adjoint orbit FIGURE 2 . Schematic showing an integral curve of the averaged Hamiltonian vector field X fl (which lies in some G-orbit G x {p}) being mapped by JG to an integral curve of Xp, (which will lie in some co-adjoint orbit O C fl*). Although one cannot show that the action variables persist as constants of the motion in the fully perturbed ( unaveraged) system, one can derive, under certain conditions, exponential bounds on their evolution of Nekhoroshev type. We turn to such estimates in the following chapters. On the other hand , the non-trivial dynamics on the co-adjoint orbits created by an F -::f 0 means that a solution curve t i------+ Xt of X fl, which must lie on some G-orbit G x {p} ~ G, moves from one T-orbit to another, in a fashion determined by F', and at a speed of order 1/E, if Eis the size of the perturbation F (see Fig. 2). These 'fast' motions correspond to those pointed out by Fasso (1995) in the context of symplectically complete isotropic foliations, which we described in the Introduction. In particular (see 4.4), one cannot guarantee that a trajectory of the averaged Hamiltonian will stay in a partial action-angle coordinate chart longer than a time of order 1/ E. This is far shorter than the exponential time scales THE AVERAGED DYNAMICS 37 one hopes to establish for the action variables (in the fully perturbed system), explaining the shortcoming of partial action-angle coordinate charts described in the Introduction. CHAPTER 5 An abstract formulation of Nekhoroshev's theorem When G = 1'n, the action-group model space G x to is just ordinary action-angle coordinates 1'n x JR n_ In this case Nekhoroshev's theorem (Nekhoroshev, 1977) asserts that for all sufficiently small perturbations F in the Hamiltonian 4.1, the action variables p evolve, in some sense, exponentially slowly. This is provided that the Hamiltonian is real-analytic and satisfies a certain 'convexity' assumption . See, e.g ., Lochak (1992) for a precise statement. Our ultimate objective is to generalize Nekhoroshev's result to the case in which G is a general compact connected Lie group. We refer to this as a problem in non-canonical perturbation theory, since the symplectic structure we on G x to is non-canonical when G is non-Abelian (in the sense given in the Introduction). Most of the techniques involved in generalizing Nekhoroshev's theorem to the noncanonical setting are not new . For example, one can still make the standard use of Lie transforms to effect symplectic coordinate changes. Since we have no desire at this point to repeat these standard arguments, it is convenient for us to abstract the setting of the theorem. The idea is that once one has certain 'basic estimates' (e.g., bounds on Poisson brackets), the proof of Nekhoroshev's theorem (although non-trivial and tedious in detail) requires only very basic properties of the underlying coordinate system and its symplectic structure. These basic estimates and other requirements are formalized as Assumptions A-C below. A generalization of Nekhoroshev's theorem, with these assumptions as hypotheses, is then stated (Theorem 5.9 below). The proof is essentially a rewriting of Lochak's proof of Nekhoroshev's theorem for conventional action-angle coordinates (Lochak, 1992; Lochak, 1993) and is relegated to Appendix A. The more significant task will be to show that 38 PRELIMINARY NOTIONS 39 Assumptions A- C indeed apply to action-group coordinates. This will be the subject of Chap. 6. Throughout this section, the reader may find it helpful to keep the example of conventional action-angle coordinates in mind. In what follows this corresponds to the choice C = "ll'n and t = !Rn. Notation Let M be a compact C 0-manifold with (possibly void) boundary whose interior int M is a complex (resp. real-analytic) manifold, and let V be a complex (resp. real-analytic) vector space. Then we denote by Av (M) the space of continuous maps u: M ➔ V that are holomorphic (resp. real-analytic) on int M. If I · I is a norm on V, then the space Av (M) is equipped with the norm llull = sup lu(x)I . xEM Preliminary notions An essential ingredient in any proof of Nekhoroshev's theorem is the complexification of the coordinates, and the analytic continuation of the Hamiltonian (which must be assumed real-analytic). The following definition formalizes in a precise way the notion that we require. 5.1 Definition Let C be a real-analytic manifold (without boundary), and suppose that there exists a nested family (co-)o~o-~ir<oo of sets (a- < a-' ⇒ co- C co-') with C 0 = C and satisfying the conditions: 1. co- is a c 0-manifold with boundary for all a- i 0. 2. int co- is a complex manifold for all a- i 0, the inclusion int co- c.....+ int Cir being holomorphic for all a- i 0. 3. Any real-analytic tensor on C 0 = C has, for some a- > 0, a holomorphic extension to int co- and a continuous extension to co-. Then we call the family (Co-)o~o-~i7 a local complexification of C. THE ABSTRACT SET-UP 40 5.2 Remark Local complexifications exist under rather general conditions, and are unique up to an appropriate equivalence (Whitney and Bruhat, 1959). See 5.6 below for an example of a local complexification in the above sense. (In Chap. 6 we shall write down a local complexification explicitly in the case that G is an arbitrary compact connected Lie group.) 5.3 Definition Let V be a real (resp. complex) normed vector space and let U C V be open. Let a real-analytic (resp. holomorphic) map h : U -+ JR (resp. C) be called (m, M)convex if there exists m, M > 0 such that D 2 h(p)(v, v) ~ ID 2 h(p)(u, v)I ~ 2 mlvl } Mlullvl Vu, v E V , Vp E U . The abstract set-up Let G be a compact real-analytic manifold (not necessarily a Lie group) and (G 17 )o:,;o-:,;o' a local complexification, with G 17 compact for all a. Lett be a finite dimensional real vector space with inner product (a, b) Na· b, and corresponding norm I · IWe assume that for some open set to C t the real-analytic manifold G x to is equipped with a symplectic two-form w. For p E to write Extend the inner product on t to a <C-bilinear form on f'C'. =t ®JR C. A norm on tlC is then given by la ® l+b ® il=Ja·a+b-b. Write B~(p) = {p' E t (C I Ip' - BR(P)I ~ p} Fix j5 E to and positive constants o- and p, and write B =Bp(f5) and BP =Bg (p), so that BP is a complex neighborhood of the real ball B. Assume that o- and pare small enough that 5.4 FUNDAMENTAL ASSUMPTIONS 41 and that w has a holomorphic extension to a (complex) bounded symplectic form on Get xBP. By bounded we mean that the induced Poisson bracket satisfies the following property: for all u and v in Ac(Gct x BP), the bracket {u,v} is assumed bounded on int(Gi't x BP), with a continuous extension to Get x BP, i.e. Consider a Hamiltonian of the form H(g, p) = h(p) + F(g, p) , 5.5 where h E AIR (B) and FE AIR (G x B). Assume that his (m' , M')-convex on int B for some positive constants m' and M', and that a and j5 are small enough that: 1. hand F have holomorphic extensions h E A<C\BP), FE AIC(Gct x BP). 2. There exists m, M > 0 such that his (m, M)-convex on int B P. Fundamental assumptions Make the following crucial assumption about the nature of the Hamiltonian vector fields defined by w: Assumption A (Existence of a 'period lattice') Let Xu denote the (complex) Hamil- tonian vector field corresponding to a function u E Ac(Gct x BP), i.e., XuJw =du.Assume that there exists a lattice {l C t, whose dimension (over Z) equals the dimension oft (over ~), with the fallowing property. Suppose WE A c (Gi'f x BP) is of the form W(g, p) = w(p) . Then if v'w(p) E vtZ = {vn I n E tZ} for some p E BP and v > 0, we require that all integral curves of Xw beginning in Ga- x {p} (a E [O, a] arbitrary) remain in Ga- x {p} for all time and are periodic with (not necessarily minimal) period 1/v. (If v'w(p) = 0, this condition implies that Get x {p} is a manifold of equilibria for X w .) 5.6 Example (action-angle coordinates) Take G = 1rn = ~n/2n-zn, t =to= ~n and w = Edqj I\ dpj, Identifying 1rn with {(z1, ... , zn) E en I lzil = 1 \fj}, we see that a local FUNDAMENTAL ASSUMPTIONS 42 G B complexification (co-)o:,;o-::;o' of G is given by co- = {(z1, ... , Zn) E en I [zj I E [1 - O', 1 + O'] \ij}, jj = ½- We satisfy Assumption A if we take t Z = 21rzn. Let k = dim t and fix a basis {,61, ... ,,Bk} for {l (which we do not require to be orthog- onal with respect to the inner product on t). Let c1 > 0 be a constant (depending only on tz, the inner product on t, and the choice of basis) such that (rj E JR) 5.7 . It is clear that such a constant exists. (It will be estimated explicitly in the case of actiongroup coordinates in Chap. 6.) Let p* E int B and O ::;; r :,;; 1 be free parameters and define the following family of (complex) neighborhoods of G x {p*} (see Fig. 1) : NEKHOROSHEV'S THEOREM 43 So that these domains are well-defined subsets of Gff x BP, the parameters p* and r must be subject to the condition 5.8 Note that Do(P*, r) = G X Brp(p*) C G X B is a real neighborhood of G x {p*} D 0 (p*, 0), and that D 1 (p, 1) = Gff x BP is the complex domain on which the Hamiltonian His defined by hypothesis. We denote the supremum norm on A<C(D.,,(p*, r)) by II· 11~•,r. The remaining hypotheses we require are: Assumption B (On 'times of validity') There exists c2 > 0, independent of m, M, "/, p* and r, such that for all u E Ac(D,,(p*, r)), 0 < 5 :s; "( and ltl :s; to = c28',05 2r/llull~*'r, every integral curve r f------t (9nPT) of Xu beginning at a point (go,Po) E D,,_s(P*,r) is welldefined and satisfies (gt,Pt) E D,,-s;2(P*,r) with IPt - Pol :s; (~~) ({) rp Assumption C (The existence of a 'Cauchy inequality' for Poisson brackets) There exist constants c3 > 0 and c4 > 0, independent of m, M, "/, p* and r, such that for all 0 < 5 :s; "/ and all u, v, WE Ac((D,,(p*, r)) with W of the form W(g,p) = w(p) one has the estimates (Recall that {u, v} = Xv J Xu J w.) Nekhoroshev's theorem Write Q(p) = '\lh(p) Et and define the 'time scales' 1 Tm=_- pm NEKHOROSHEV'S THEOREM Tn =(sup 1n(p)l)- 44 1 Th'"= (,o2sup sup ID 3 h(p)(u,v,w)l)-I pEBP pEB u,v,wEl and the associated 'dimensionless constants' We allow Th"' = oo (in which case c5 = oo). A constant with the physical dimensions of the Hamiltonian Ho is Recall that k denotes the dimension of the real vector space t. 5.9 Theorem (Nekhoroshev-Lochak (abstract form)) Consider the scenario desribed in this section with Hamiltonian H of the form 5.5, under the Assumptions A, Band C above. Let II • JI denote the supremum norm on AC(Gir x B 15 ) and define the 'perturbation parameter ' Then there exist positive constants a = a(k), b = b(k) , c, Eo = Eo(k, c1, ... , c1) , to = t0 ( c1, ... , c7) and r 0 = ro (c1, ... , C7), such that for all E ~ Eo every (real) solution curve tr-+ (gt, Pt) E G x B of XH with p0 = p obeys the exponential stability estimate ltl < t exp(cE-a) -Tn "" o ⇒ IPt - Pol b P - ""< roE • A proof of 5.9, based largely on Lochak (1992) and Lochak (1993), is outlined in Appendix A. Estimates for the constants a, b, c, Eo, to, r 0 obtained in Appendix A are summarized below. For pedagogical reasons, we have not attempted to optimize the exponents a and b (these are optimized in Lochak and Neishtadt (1992) and Lochak (1993)). Preliminary investigations of ours suggest that a= 21k and b = /k should be possible (c.f. a= 2~ and b = 2~ in op. cit, n the degrees of freedom). For more information see Appendix A. NEKHOROSHEV'S THEOREM 5.10 Addendum In Theorem 5.9 above one may take 1 a=--- 4(1+k) ' 1 b = 2(1 + k) ' C = 1 where 2 l1 } =max { 1, 0192a , (l)k+l/ l 4 t_ C7 = y'S- 2 C 2 7 , 45 CHAPTER 6 Applying the general theorem to action-group coordinates In this section we show that the abstract formulation of Nekhoroshev's theorem given in the previous section applies to action-group coordinates (Proposition 6.1 below). The corresponding estimates will be combined with the local existence theorem for action-group coordinates (Theorem 3.10) to deduce Nekhoroshev estimates for momentum maps in Chap. 7. 6.1 Proposition Nekhoroshev's theorem, as formulated in 5.9, applies in the case that G is a compact connected Lie group and where t, to and w = wa have the meanings given in Chap. 2. Recall (see, e.g., Brocker and tom Dieck (1985, V8.l)) that up to a finite covering any compact connected Lie group is the direct sum of a semisimple and Abelian (toral) factor. Although 6.1 holds for any compact connected Lie group G, we shall only prove it in the case that G is semisimple. The Abelian case is already covered by Nekhoroshev's original statement. Furthermore, it turns out that there is essentially no 'coupling' between the Abelian and semisimple factors in the general case. To simultaneously keep track of both an Abelian and semisimple factor just introduces uninstructive bookkeeping to the calculations. We leave these for the more energetic reader to work out. For the remainder of this section G denotes a compact connected semisimple Lie group. In this section we shall make use of basic properties of semisimple Lie algebras, as well as their root and inverse root systems. We refer the reader to Brocker and tom Dieck (1985) for a discussion of the relationship between these roots and the Wey! chambers, and for further background. Here a root will always mean a real root, in the sense of, e.g., Vl.3, op. cit. 46 THE LOCAL COMPLEXIFICATION OF G 47 The local complexification of G It is now convenient to fix the choice of Ad-invariant inner product on fl introduced in Chap. 1, which until now has been completely arbitrary. Define (~,T/Efl) · Minus one times this product is known as the Killing form of fli it is symmetric, Adinvariant, and known to be nondegenerate precisely when fl is semisimple. Remember that we have been denoting the unique C-bilinear extension of our product to fie by~ •'f/ also. We continue to denote the norm on fl induced by the inner product (and its obvious extension Recall that we assume that G is realized as a subgroup of SO(na, Ill), so that we may view fie as a subalgebra of enc xna . The latter Lie algebra is equipped with a natural symmetric C-bilinear form, namely the Schur-Hadamard product S(A, B) 1 =-Trace(ABT) na An associated norm on enc Xna is given by IAls , =JS(A, A) = -na1 Trace(AAT) where a bar denotes complex conjugation. Note that as G C SO(na, Ill), (g E G) 6.2 and 6.3 lgAls = IAls = IAgls THE LOCAL COMPLEXIFICATION OF 0 48 By writing things out with respect to the standard basis of cn,a and applying the Schwartz inequality several times, one can derive the estimate IABls ~ vnalAlslBls . 6.4 If if > 0 is chosen sufficiently small, then a well-defined local complexification (G")o:::;a:::;ir of G is given by 6.5 It will be convenient to assume We need to establish the relation between I · I (the norm on gC determined by the Killing form) and the restriction to gC of I · Is- Suppose that A, B E ((;1-G xna and g E G. Then since Ad 9 A= gAg- 1 , 1 S(Ad 9 A, Ad9 B) = -Trace(gAg- 1 (g- 1 BT gT) no 1 = -Trace(gABT g- 1 ) (since g E SO(no, JR)) no 1 = - Trace(ABT) (since Trace is invariant under conjugation) no f = S(A, B) . This shows that S ( ·, ·) is invariant with respect to the adjoint action of G. In particular its restriction to g is Ad-invariant. But as G is semisimple, there is up to a constant only one Ad-invariant inner product on g, namely the Killing form (see op. cit.). By the uniqueness of extensions of JR-bilinear forms on g to C-bilinear forms on gC, it follows that 6.6 for some k1 > 0. The constant k1 depends on the realization of G C SO (no, JR) and is the only constant we leave implicitly defined. It is easily computed on a case-by-case basis. EXISTENCE OF A PERIOD LATTICE (ASSUMPTION A) 49 From 6.6 it follows that l~ls = Jki°l~I 6.7 In particular we deduce from 6.4 and 6.7 the estimate 6.8 where 6.9 Existence of a period lattice (Assumption A) Define k =(dimG - rankG)/2 ~ k =dim t = rankG Then with respect to the fixed chamber to, fl has k positive real roots We assume that these have been ordered so that {a 1, ... , ak} is the basis of t* consisting of the indecomposable elements of { a 1, ... , a;;;}. The corresponding inverse roots will be denoted ai, . .. , at E t. They are given by 6.10 where K,: t-+ t* is the isomorphism (K,(a) , b) =a ·b, and B( ·, ·) is the induced inner product on t*. It is a fact that the inverse roots lie on the integral lattice I of T in t (defined by 4.2). Suppose W E AIC(G5 x B P) is of the form W(g,p) = w(p). Then by the equations of motion 2.11, an integral curve t t-+ (gt, Pt) of X w beginning in ca x {p0} (0 ~ a ~ er, p0 E BP) is given by (gt,Pt) = (goexp(tv'w(po)),Po), Suppose that there exists av > 0 ESTIMATING c 1 50 such that Vw(p) E vl. In that case tVw(po) lies in t (i.e., is rea0 for any t E JR so that exp(tVw(po)) EC. Now Igo - g~ls ~ a for some g~ EC (as go E C 0"). By 6.3, we have lgoexp(tVw(po)) - g~exp(tVw(po))ls = Igo - g~ls ~ a . Since g~exp(tVw(pO)) E C, this shows that g0 exp(tVw(po)) E ca for all t E JR, i.e., (gt, Pt) E ca x {po} for all t E JR. Furthermore, as Vw(po) E vl, this solution curve has 1/v as a period. We therefore satisfy Assumption A of Chap. 5 if we take tZ to be any k-dimensional sublattice of I. While the 'optimal' choice for tZ is I itself, future computations are simplified by choosing tZ to the sublattice generated by the inverse roots: Note that tZ = I if C is simply connected. Estimating c1 A basis {,8 1 , ... , ,Bk} for tZ is given by ,Bj =a;. We will estimate the constant c of 5. 7 1 in terms the Cartan integers nij of the root system . These are defined by (1 ~ i,j ~ k) . It is a fact that the restriction to t of -1 times the Killing form can be expressed in terms of the positive real roots according to k a· b = 81r 2 L (am, a)(am, b) (a, b E t) m=l It follows from 6.10 that 6.11 Let Ai, ... , Ak > O denote the eigenvalues of the (symmetric and positive definite) matrix (mijh~i,j~k· Then for some orthogonal transformation A : JRk ➔ JRk and any r = BOUNDS ON >.p 51 k h/31 + •••+ Tk/3kl = L TimijTj 2 i,j=l k = L(AT)jAj(AT)j j=l k k ~ (sup>.j) 2)AT)J = (supAj) J j=l J L hl 2 j=l ~ (sup Aj)k(sup 1Tjl) 2 • J J We therefore satisfy 5.7 if we choose c1 to be the square root of the maximum eigenvalue of the matrix (mij), which is defined in terms of the Cartan integers by 6.11. 6.12 Example G = SU(3) (see Fig. 1). In this case{?_;= I, k = 2 and k = 3. The matrix of Cartan integers is 2 -1 1 1 1 2 so that The largest eigenvalue of the right-hand side is 72n2, so that for G = SU(3) we may take C1 = 61rv'2. Bounds on Ap Recall (Chap. 2) that Ap (p E cy) is defined as the inverse of adp : t1-C ➔ t1-C_ Since Ap appears in the equations of motion 2.11 and in the formula for the Poisson bracket 2.15, we will need to estimate the operator norm of Ap before proceeding to verify Assumptions Band C. BOUNDS ON >.p 52 a*1 FIGURE l. The system of inverse roots for G = SU(3) (k = 2, k = 3). Here 0 indicates a positive inverse root, and @ a negative inverse root. Arrows indicate the location of the basis roots. The shaded region indicates the Wey! chamber to. 6.13 Remark Since the operator adp : t..LIC --+ t..LIC becomes singular as p approaches the walls 8to of to, the norm of Ap is unbounded as p --+ 8to. We will see later that this results in a deterioration in the estimates of Assumptions B and C as p--+ 8to. This phenomenon is particular to the non-Abelian case. Fix p E ~- Note that dim t..LIC = 2k. For 1 ~ j ~ k let ~j (resp. ~-j) denote an element of t..LIC spanning the weight space corresponding to the root ltj (resp. -aj). Then {Ck, ... ,~-1,6, ... ,~x;} is a basis for t..LIC (over C), and (Here the roots aj have been extended to C-linear functionals on tic.) So with respect to the above basis, adp diagonalizes with diagonal BOUNDS ON >.p 53 Consequently Ap = ad; 1 diagonalizes with diagonal -1 -1 1 1 ) ( 21ri ((X1c,, p) ' ••• ' 21ri (0!1, p) ' 21ri (0!1, p) ' ••• 21ri (O!k, p) It follows that for any ~ E t1-C, Since (cxj, a) E ~ for all a E t, I(O!j, p) I ~ I(o:j, Rep) I- Also, every positive root O!j (1 ~ j ~ k) can be written as a linear combination of the basis roots o: 1 , ... , O!k with positive coefficients. The above estimate can therefore be written 6.14 Manipulating 6.10, one shows that 2A:( cxJ) 6.15 O!j = lo:; 12 Recall that p E ~- The diagonal matrix representation of adp above, Equation 2.7, and the definition of ~ show that RepEto. If a E to, then an elementary geometric property of root systems is (see Fig. 1) that So for any j with 1 ~ j ~ k one has o:J Ia - ato I ~ Io:; I •a = (A:- 1 ( o:J), a) Io:; I 1 * = 2Io:j I(o: j , a) , using 6.15. This shows that (a E to, 1 ~ j ~ k) . In particular we can apply this to 6.14, with a= Rep, to conclude that BOUNDS ON DIRECTIONAL DERIVATIVES 54 i.e., 6.16 where 6.17 Note that k3 can be expressed purely in terms of Cartan integers: k§ is the maximum diagonal entry in the matrix (mijh(i,j(k defined by 6.11. For example if G = SU(3), then by 6.12, k 3 = 41ry13. Bounds on directional derivatives In the remainder of this section the reader may find it helpful to refer frequently to Fig. 1. In the notation of Chap. 5, we now prove: 6.18 Lemma Let u E A<C (D.y(P*, r)), ( E 1:f and r E t<C be given. If O < o ~ 1 , then for all (g,p) E D"l_s(p*, r), 1. I ( . ~;(g,p)I ~ 2k2(e -1) 1(111:!~•,r ' 2. IT. ~;(g,p)I ~ lrll~~~~•,r PROOF. Fix (g,p) E D"l_s(P*, r). By definition where J(t) = u(gexp(t(),p). We seek v > 0 sufficiently small that !ti~ v ensures that 6.19 Then !ti ~ v ensures that J(t) E <C is well-defined , and by Cauchy's inequality for a holomorphic function of a complex variable, 6.20 (t E <C) . BOUNDS ON DIRECTIONAL DERIVATIVES 55 Since (g, p) E D'Y-o (p*, r) (and because G is com pact) there exists gIR E G such that 6.21 We have lg exp(t~) - Gls ::;; lg exp(tO - g!R ls ::;; lg exp(t~) - gls + jg - g!R js ::;; Jnclgls Iexp(t~) - idls + jg - g!R ls by 6.4 ::;; Jnc( jg!R js + jg - g!R js) I exp(t~) - idjs + jg - g!R js ::;; Jnc( 1 + (1 - 6)a) I exp(tO - idjs + (1 - 6)a 6.22 ::;; 2Jncl exp(t~) - idls + (1 - 6)a by 6.2 and 6.21 Since a::;; 1 Combining the Taylor series expansion of exp(t~) E cna xna with 6.4, 6.7 and 6.9: assuming that 6.23 So (assuming 6.23) 6.22 implies Therefore lg exp(t~) - Gls ::;; 1 a (so that 6.19 holds) provided 6.24 TIMES OF VALIDITY OF HAMILTONIAN FLOWS (ASSUMPTION B) 56 In summary, 6.19 holds if 6.23 and 6.24 do. Since 6.24 is stronger than 6.23 (o, o- ~ 1), we may take With this choice, 6.20 becomes 6.18.1, the first estimate of the lemma. The proof of 6.18.2 is simpler and left to the reader. □ We write oullp.,r l -og = 'Y - sup IOU -~ ~ (g,p)ED ..,.(p.,r) og I ' • One has An analogous statement holds for I(ou/ op) (g , p) I- The following corollary of 6.18 is therefore clear. 6.25 Corollary Let u E Ac(D,.y(p*, r)) be given and suppose O < o ~ 1 . Then 1. 2. Times of validity of Hamiltonian flows (Assumption B) 6.26 Lemma (Bounds on the components of a Hamiltonian vector field) Let u E Ac(D-y(P*, r)) be given and suppose O < o ~ 1 . Then 1. 2. Recall that ~u and Tu are defined by 2.9 and 2.10. Note also that as B2p is closed , 5.4 ensures IB2 p - otol > 0. TIMES OF VALIDITY OF HAMILTONIAN FLOWS (ASSUMPTION B) 57 PROOF . The definition 2.9 of ~u gives, with the help of 6.16, {)u l p.,r ll~u ll~~~ ~ 1 op -y- S + k3 ( pEB~}!!kr,(p.) IRep- )-l 1 l p.,r {)u {)g -y- S oto l Recalling that 5.8 ensures that B~rS)ri5 C BP (0 ~ 1 ~ 1, 0 < o ~ 1 ), we have inf IRe p - ato1 )-l ( pEB~ro)rr,(p.) ~ IB2p - ato1 - 1 . Applying 6.25, and exploiting the fact that r ~ 1, one easily deduces 6.26. 1. The inequality 6.26.2 follows similarly from 2.10 and 6.25.1. □ We are now ready to turn to the verification of Assumption B. Let u E A«\D-y(P*, r)) be given and suppose that O < 8 ~,.Lett f-7 (9t,Pt) be an integral curve of Xu, and suppose that the initial condition satisfies (go,Po) E D-y-S(P*, r), i.e., ('Y-S)ri5( ) Po E B rp P* • 9o E ch-S)a According to the equations of motion 2.11, 6.27 6.28 Because D-y-s; 2 (p*, r) is compact, the solution (gt, Pt) E D-y- s; 2 (p*, r) is well-defined until it reaches the boundary of D-y- s; 2 (p*, r) . That is, there exists a time sto (s = ±1, 0 < t 0 ~ oo) such that 6.29 ⇒ and such that at least one of the following holds: 6.30 l9sto - G ls = b- o/2)a- , 6.31 According to 6.27, rsto 9sto - 90 = Jo 9t~u(9t, Pt)dt • TIMES OF VALIDITY OF HAMILTONIAN FLOWS (ASSUMPTION B) 58 From 6.29 and 6.8 we deduce by 6.2 Applying 6.26.1: where 6.32 6.33 Remark Notice that ( --+ oo as the real ball B 2 p = B 2 p(j5) approaches the walls of fa. Since g0 E ab-S)a, there exists g~ E G such that [g0 - g~[s ~ (, - 5)a-. We compute So, supposing 6.30 holds , we have 6.34 According to 6.28 {8to Pst 0 =Po+ Jo Tu(gt,Pt)dt From 6.29 and 6.26.2 (with 5 replaced by 5/2) we deduce INEQUALITIES FOR POISSON BRACKETS (ASSUMPTION C) 59 compute IPsto - Brpl ~ IPsto - P~I ~ IPsto - Pol + !Po - P~I So, supposing 6.31 holds , we have 2 ) ?fp5 r 1 6.35 to ~ ( 8k2(e - 1) llull~•,r Since 6.30 and/or 6.31 holds, 6.34 and/or 6.35 holds. Therefore if we take 1 6.36 c2=------ - 8k2 max{(, e - 1} Arguments like the above also show that !Pt - Pol~ (~~) (~) rp With the above choice of c2 , Assumption B of Chap. 5 is thus established. Inequalities for Poisson brackets (Assumption C) Let u, v E A<C(D,,(p., r)) be given and suppose O < 5 ~ 1 . Fixing some (g,p) E D,,_o(p.,r), we have {u, v}(g,p) = (du,Xv(g,p)) = (v(g,p) au 0 au Og (g,p) + Tv(g,p). Op (g,p) = J'(O) ' where J(t) =u(gexp(t(v(g,p)),p+ trv(g,p)) . by 2.8 INEQUALITIES FOR POISSON BRACKETS (ASSUMPTION C) 60 We seek v > 0 sufficiently small that [t[ :::;; v guarantees 6.37 Then [ti :::;; v ensures that J(t) E C is well-defined and, by Cauchy's inequality, 6.38 1 JJ uJJP•,r [{ u,v}(g,p)[:::;; - sup [J(t) J:::;; 'Y V ltl~v V To satisfy 6.37 we need Jg exp(t~v(g,p)) - G[s:::;; 1 CJ 6.39 6.40 and [(p + trv (g, P)) - Br i5 (p*) [ :::;; 1r P • Arguing as in the proof of 6.18.1, we satisfy 6.39 if Since (g ,p) E D-y-s(p*, r), it suffices, by 6.26.1, to ensure 6.41 ap8 2 r Jt[ :::;; 2ki(e - 1)( J[v[J~•,r ' where ( is the constant defined by 6.32. Arguing similarly, one sees that 6.40 holds provided r8p Jtl :::;; ITv (g,p )I By 6.26.2, it suffices to ensure 6.42 To summarize, if 6.41 and 6.42 hold, then so do 6.39 and 6.40, and hence 6.37. It follows that we may take With this choice of v, 6.38 becomes INEQUALITIES FOR POISSON BRACKETS (ASSUMPTION C) 61 Since (g, p) E D-y-8(P*, r) was arbitrary, the first estimate of Assumption C (Chap. 6) holds if we take 6.43 If WE Ac(D,.(p*, r)) is of the form W(g,p) = w(p), then (by, e.g., 2.15), 8u { u, W}(g, p) = og (g, p) • v'w(p) = J'(O) , where J(t) = u(gexp(tv'k(p)),p). Cauchy's ineq uality and the by now familiar argument leads to the estimate r ll{u, W}II~·· :::; 2k2( e - 1) O(J sup lv'w(p)I . B(-y-o)r,o( ) PE r,o P• Since o :=:; 1, the second estimate of Assumption C certainly holds if we take 6.44 This completes the proof of Proposition 6.1 in the case of a semisimple G. CHAPTER 7 Nekhoroshev-type estimates for momentum maps In this section we combine Theorem 3.10, Theorem 5.9 and Proposition 6.1 to obtain the Nekhoroshev type estimates on momentum maps discussed in the Introduction. Let G be a compact connected Lie group and (P, w, G, J) a Hamiltonian G-space. Consider a Hamiltonian of the form H= Ho+F, where Ho is G-invariant, and F is arbitrary. Existence of action-group coordinates. Assume that the hypotheses of Theorem 3.10 apply (working 'locally' ala Remark 3 .1 1 if necessary), so that there exists a diffeomorphism </>: G x U ➔ P (U = <p- 1 (J(P)) n to) such that (Hoo </>)(g,p) = h(p), for some smooth h : U ➔ JR. Recall that <p : fl ➔ fl * is the isomorphism corresponding to the fixed Adinvariant inner product on fl. Complexification of the coordinates. Fix some x E P and define (g,p) = ¢- 1 (x) . Fix a positive constant p small enough that the real closed ball B 2 p(f5) is contained in U and the complex closed ball B;(p) (defined in Chap. 5) is contained int~ (defined in Chap. 2). Fix a positive constant a- ~ 1 small enough that (Go-)o,,;;o-,,;;a, as defined by 6.5, is a local complexification of G in the sense of 5.1. Analyticity of the Hamiltonian. Assume that h E AIR(Bp(p)) and Fo</> E AIR(G x Bp(p)) (notation as in Chap. 5) . Furthermore assume that a- and pare sufficiently small that h and F o </> have holomorphic extensions h E A<C(Bg(p)) and F o </> E A<C(Ga x Bg(p)). Convexity of the unperturbed Hamiltonian. Assume that his (m, M)-convex on int B: (p) in the sense of 5.3, for some m, M > 0. 62 7. NEKHOROSHEV-TYPE ESTIMATES FOR MOMENTUM MAPS 63 Constants. Write Q(p) = 'vh(p) Et and define the 'time scales' 1 Tm=-_pm Tn =( sup ID(p)l)-l 1 TM=- j5M Th'"= (j5 2 sup sup ID 3 h(p)(u, v, w)l)-l pEBP pEB u,v,wEl and the associated 'dimensionless constants' Recall that we allow Th"' = oo. The perturbation parameter. Let II · II denote the supremum norm on .A<C(GB' x BP) and define the 'perturbation parameter' where E = ffj5/TM. We have the following corollary of Theorem 5.9 and Proposition 6.1. (Recall that k denotes the rank of G.) 7.1 Corollary There exist positive constants c 1 , c2 , c3 , c4 independent of h and the pertur- bation F, and positive constants a= a(k), b = b(k), c = c(c1, ... , c7), Eo = Eo(k, c1, ... , c7), to= to(c1, ... , c7) and ro = ro(c1, ... , c7), such that for all E ~ Eo and every integral curve tH Xt E P of XH with xo = x one has where O C _g* is the co-adjoint orbit through the point J(x 0 ). Estimates for the constants a, b, c, Eo, to, ro (which are not optimal) appear in Addendum 5.10. If G is semisimple (and one takes -1 times the Killing form as the fixed inner product on _g), then c 1 depends only on the Cartan integers of G (according to the explicit computation 7. NEKHOROSHEV-TYPE ESTIMATES FOR MOMENTUM MAPS 64 presented in Chap. 6) and one may take 1 c2=------- 8k2 max{(, e - 1} c3 = 2k2(e - 1) max{l, (} c4 = 2k2(e - 1) ( = 1 + 2k2k3(e - l)p [B2p(p) - oto[o- The constant k3 (see 6.17) depends only on the Cartan integers. The constant k 2 (see 6.9) depends on the realization of G as a linear group. The reader should keep Remark 6.33 in mind. PROOF. That one arrives at the following estimate for the action variables is clear: compute [J(xt) - O[:::; IJ(xt) - 9t9o 1 • J(xo)I = l9t • <p- 1 (Pt) - 9t • <p- 1 = l<p- 1 (Pt) - <p- 1 (Po)I = [Pt - Po [ (Po)I , since the isomorphism <p : t -+ i is isometric. This proves that the estimate claimed in the corollary indeed holds. □ Part 2 Geometry Outline of Part 2 Here we describe geometric constructions underlying the analyses given in 'Part 1: Dynamics.' We assume that the reader has at least read the Introduction and the historical remarks at the end of Chap. 3. The exposition is otherwise relatively self-contained. In Chap. 8 we study a special class of Hamiltonian G-spaces, namely those possessing points of 'regular co-adjoint orbit type.' It is shown (Theorem 8.14) that essentially all geometric information describing such a space is encoded in a symplectic cross section, which is a lower dimensional Hamiltonian T-space, where T C G denotes some maximal torus (we assume that G is compact and connected). This result may be viewed as an application of the philosophy of Guillemin and Sternberg (1984, §41) (see also Guillemin et al. (1996)). In Chap. 9 we show how action-group coordinates, a generalization of action-angle coordinates that was applied to perturbation theory in Part 1, can be realized as a symplectic submanifold of T*G. We derive explicit expressions for the symplectic structure and Hamiltonian vector fields in these coordinates. In Chap. 10 we use the results of Chap. 8 to reduce the problem of constructing actiongroup coordinates in an integrable system, to the construction of action-angle coordinates in an associated symplectic cross section (Theorem 10.2). We tackle the problem of global existence by relating work of Duistermaat (1980). We also prove 'semi-global' results, which are more convenient to apply in concrete examples. This includes a proof of Theorem 3.10 of Part 1 (here stated as Corollary 10.12). In Chap. 11 we apply the preceding theory by constructing explicit action-group coordinates in the axisymmetric Euler-Poinsot rigid body. The constructions of Chap. 10 apply to systems that have zero dimensional MarsdenWeinstein reduced spaces, with respect to some known symmetry group G. In Chap. 12 we show how one can sometimes enlarge G in a system with two dimensional reduced spaces to G' = G x 5 1 , in such a way as to render the G'-reduced spaces zero dimensional. We demonstrate this in the case of the rigid body in Appendix D. 66 OUTLINE OF PART 2 67 In Appendix E we include a detailed proof of Weinstein's Symplectic Correspondence Theorem, which we need in Chap . 12 . CONTENTS OF PART 2 Chapter 8. On Hamiltonian G-spaces with regular momenta 68 Chapter 9. Action-group coordinates as a symplectic cross section 79 Chapter 10. Constructing action-group coordinates 86 Chapter 11. The axisymmetric Euler-Poinsot rigid body 97 Chapter 12 . Converting dynamic integrability to geometric integrability 104 Chapter 13. Concluding Remarks 117 Appendices (Parts 1 and 2) 119 CHAPTER 8 On Hamiltonian G-spaces with regular momenta Preliminaries Before turning to our main subject, we review and supplement terminology used in Chap. 3 of Part 1. We already assume some familiarity with the structure of compact Lie groups, say with what has been summarized in Chap. 1 of Part 1. We work throughout in the category of smooth manifolds and maps, where 'smooth' means C 00 or real-analytic. If a Lie group G acts on a symplectic manifold (P, w) by symplectic diffeomorphisms, and the infinitesimal generators EP (E E fl) are global Hamiltonian vector fields, then one says that G is acting in a Hamiltonian fashion, or that the action is Hamiltonian. In that case there exists a map J : P -+ 9*, called a momentum map, with the property that it delivers Hamiltonian functions Je : P -+ JR for the generators according to the formula Je( x) = (J (x), E). So, if we denote the Hamiltonian vector field corresponding to a Hamiltonian H by XH, we have EP = XJc If furthermore the momentum map is Gequivariant (with G acting on fl* by the co-adjoint action), then we shall refer to (P, w, G, J) (or simply P) as a Hamiltonian G-space. For background on Hamiltonian actions and momentum maps see Marsden and Ratiu (1994), Abraham and Marsden (1978, Chapter 4) or Guillemin and Sternberg (1984). We limit ourselves throughout to the case of G compact and connected. By the co-adjoint orbit type of a point x E P we shall mean the orbit type of J(x) E 9*. The space P has uniform co-adjoint orbit type (T) (T C G denoting a maximal torus) if and only if J(P) C fl;eg· Here fl;eg denotes the regular points of the co-adjoint action ( not to be confused with regular values of J). In that case we will call P a Hamiltonian G-space with regular momenta. Our interest in such spaces is motivated by the fact that, under an appropriate integrability hypothesis, they admit action-group coordinates which, 68 PRELIMINARIES 69 as we showed in Part 1, can be used as a geometric framework for perturbation theory. A simple example is geodesic motions on S 2 (provided we exclude the trivial motions); see, e.g., Chap. 3 of Part 1. For us the construction of 'coordinates' in a Hamiltonian system will mean the construction of an appropriate equivalence between Hamiltonian G-spaces: 8.1 Definition Two Hamiltonian G-spaces (P ,w ,G,J 1 1 1 ) and (P ,w ,G,J 2 2 2 ) will be said to be equivalent if there exists a G-equivariant symplectic diffeomorphism </>: P 1 -+ P2 such that J 2 o </> = J 1. Henceforth T C G denotes a fixed maximal torus, t C g its Lie algebra, and W C i =Ann[g, t] a fixed (open) Wey] chamber in g* (terminology and notation as in Chap. 1 of Part 1); (J will denote a fixed regular co-adjoint orbit, and µ 0 the unique point of intersection of (J with W. The natural projection g*-+ t* (dual map of inclusion) restricts to an isomorphism i: !-+ t*, which identifies W with an open set fo =i(W) . We refer to W as a Wey! chamber also. We now turn to a systematic study of Hamiltonian G-spaces with regular momenta. While we make no explicit assumptions on the orbit type (Gx) of points x E P, the following is worth noting. 8.2 Proposition If P is a Hamiltonian G-space with regular momenta, then dim Gx ~ rankG for all x E P. This follows from the equivariance of J and the identity 8.3 which follows from the definition of a momentum map. For any Hamiltonian G-space P, the open G-invariant subset P' = J- 1 (g;eg) is a Hamiltonian G-space with regular momenta. If, however, dimGx > rankG for all x E P, then ON THE GEOMETRY OF THE CO-ADJOINT ACTION 70 Proposition 8.2 implies that P' is empty. The following example (relevant to the so-called 1:1:1 resonance) is a case in point. 8.4 Example Equip P =lR. with the standard symplectic structure, and let G =SU(3) act 6 (faithfully) on P by identifying JR 6 with C3. Then G acts symplectically, as is well-known, and so admits a momentum map, on account of the Poincare lemma. This momentum map is equivariant since SU(3) is semisimple (Guillemin and Sternberg, 1984). One finds that for all x E P, dim Gx 2'. 3 >rank G = 2, and that accordingly J(P) nfl;eg = 0. (In the 1:1:1 resonance one studies cubic and higher order perturbations of the G-invariant Hamiltonian H(z1, z2, z3) = ½lz11 2 + ½lz21 2 + ½lz31 2-) Henceforth (P, w, G, J) denotes a Hamiltonian G-space with J(P) C fl;eg· On the geometry of the co-adjoint action Our first observation is that there exists a natural G-equivariant trivialization fl;eg ~ 0 x W (with G acting on Ox W according tog · (v, w) = (g • 11, w)). This trivialization is given by µ M (1ro(µ), 1rw(µ)), where 1ro : fl;eg ----+ 0 and 1rw : fl;eg ----+ W are defined implicitly by =g • µo 1rw(g • = 1ro(g • w) 8.5 w) (g E G, w E W) , w every element of fl;eg being of the form g • w for some g E G and w E W. Indeed W is a global slice at µ 0 for the co-adjoint action of G on fl;eg, so that there exists a natural G-equivariant isomorphism fl;eg ~ G Xy W (see Appendix B). Since all points in W have isotropy group T, we have G Xy W ~ G/T X W ~ 0 x W. Note that the fibers of 1ro : fl;eg ----+ 0 are the Weyl chambers in fl*; the fibers of 1rw : fl;eg ----+ W are the regular co-adjoint orbits. 8.6 Example Take G = S0(3) and let T C G be the rotations about the e3 axis. The Lie algebra fl = .so(3) can be identified with JR 3 via the isomorphism ~ Mt : JR 3 ----+ .so(3) defined by tu = ~ x u (u E JR 3). We have a corresponding identification fl* ~ JR 3. The THE FIBERING OF P BY HAMILTONIAN T-SPACES 71 co-adjoint action is then represented by the standard action of SO(3) on R 3. We have i = span{e3}. For a Weyl chambering* choose W ={te3 I t > 0}. For a regular co-adjoint orbit O choose the unit sphere S 2 C R 3. The unique intersection point of W and O is =e3. The projection rro : R \{0} ----+ S is given by rro(µ) =µ/ 11µ11 - The projection rrw: R 3\{0}----+ W ~ (0, oo) is given by rrw(µ) =llµII3 µo 2 Since J (P) C g;eg, it is natural to pull the projections rro and rrw back using the momentum map: Define = J : p ----+ 0 , j =i rrw J : P ----+ to t* 7r 8.7 7rO o o o C In the following paragraphs we shall see that each fiber of 7r encodes the geometric structure of the full space P. Later, we shall see that j : P----+ t* is the momentum map for a natural Hamiltonian action of Ton P. Unlike the action of T as a subgroup of G, this new T-action leaves the fibers of 7r invariant. Furthermore the orbits of this action do not depend on the choice of maximal torus T . The fibering of P by Hamiltonian T-spaces The map 7r is G-equivariant. Since G acts transitively on 0, it follows that 1r : P----+ 0 is a surjective submersion. 8.8 Theorem (Guillemin-Sternberg) Each fiber Fv = (v) of is a symplectic sub1r- 1 1r manifold of P invariant with respect to the action of the maximal torus Gv, Furthermore, Gv acts on Fv in a Hamiltonian fashion (with respect to the symplectic structure Wv with an equivariant momentum map JV : Fv ----+ g: given by JV ( x) =wlFv) =J (x) lf1v (x E Fv) . This makes Fv a Hamiltonian Gv-space (Fv, Wv , Gv, JV). Each element g E G maps each fiber of 1r symplectomorphically onto another fiber, and this induced action on the fibers is transitive. THE FIBERING OF P BY HAMILTONIAN T-SPACES 72 Theorem 8.8 follows, for example, from the symplectic cross section theorem (Guillemin and Sternberg, 1984, §41). For a statement of this theorem the reader is referred to Guillemin et al. (1996), from whom we also borrow the following terminology: 8.9 Definition Write F = Fµ wp =wµ and JF = 0 , 0 Jµo. The Hamiltonian T-space (F, wp, T, JF) will be referred to as the symplectic cross section of (P, w, G, J). Note that JF = i oJIF. In fact, the symplectic cross section theorem gives more information than is contained m the statement of Theorem 8.8. It shows that 1r : P -+ 0 is naturally isomorphic to the bundle G x T F -+ G /T. In particular, 1r : P -+ 0 is a symplectic fibration, meaning that it is a locally trivial fiber bundle, with fibers modeled on F, whose structure group is contained in the symplectomorphism group of F. The symplectic cross section theorem also gives a characterization of the symplectic connection on 7r : P -+ 0 (see 8.11 below for a definition). This characterization allows one to recover the symplectic structure won P from: (i) its restriction to F, and (ii) the natural co-adjoint orbit symplectic structure of the base 0. The interested reader is referred to op. cit. for details. For our purposes, Theorem 8.8 and the rather straightforward observation in Lemma 8.12 below will suffice. Since all maximal tori of G are conjugate, we can replace the action of Gv on a fiber Fv (v E 0) by a Hamiltonian action of the fixed torus T = G µo . (In fact, a slightly stronger statement is true; see Corollary 8.18.) This makes each fiber a Hamiltonian Tspace. By Theorem 8.8 there exists for any fibers Fv 1 and Fv2 an element of G mapping Fv 1 symplectomorphically onto Fv2 • This map is in fact an equivalence of Hamiltonian T-spaces, in the sense of 8.1. In particular, each fiber of 1r is equivalent as a Hamiltonian T-space to the symplectic cross section. 8.10 Remark (On G-invariant dynamics) If H : P-+ IR is a G-invariant Hamiltonian, then the fibers of 1r = 1ro o J will be XH-invariant , by Noether's theorem (which states that the preimage of any set under the momentum map is XH-invariant) . Furthermore, THE SYMPLECTIC CONNECTION 73 the dynamics in any two fibers will be conjugate because any fiber can be mapped symplectomorphically onto any other by a group element. So any feature of the dynamics in a given fiber is repeated in all the fibers. For example, the existence of an equilibrium point for XH in fact implies the existence of an entire manifold of equilibria diffeomorphic to 0. Borrowing language from perturbation theory (where one is interested in non-invariant perturbations to H), we say that such a Hamiltonian is more 'resonant 1 .' This built-in resonance is particular to the non-Abelian case, for otherwise O is just a point and 1r has only one fiber, viz. P. The symplectic connection 8.11 Definition The symplectic connection on 1r : P ➔ 0 is the assignment to each x E P of a space Horx = (kerTx1r)w called the horizontal space at x which, on account of F1r( x ) C P being a sym plectic su bmanifold, is complementary to ker T x1r in T xP. The restriction of w to the horizontal spaces is easily characterized: 8.12 Lemma 1. Horx C Tx(G • x) 2. w(~p(x), 77p(x)) = (J(x), [~ , 77]) PROOF. Since 1r (x E p) . (xEP; (,77Eg) = 1ro o J , we have ker T x1r :J ker T xJ . Taking w-orthogonal comple- ments, we obtain Horx C (kerTxJ)w. By the definition of a momentum map, g* ? P ➔ P/G has the 'dual pair' property, i.e. (kerTxJ)w = Tx(G • x) . So 1 holds. To prove 2, we observe that as J is equivariant, it is also infinitesimally equivariant, from which it follows that {Je,J11 } = J[e, 11 ], where {J,h} = XhJ X1Jw (see, e.g., Abraham and Marsden (1978, Theorem 4.2 .8) for a proof). We can now compute w(fr(x), 77p(x)) = w(X1e(x), X177 (x)) = {Je, J 11 }(x) = J[e,111(x) = (J(x), [~, 77]) · □ 1 A perturbation theorist might also coin the term 'degenerate.' From the symmetry point of view, this terminology is misleading, as the phenomenon we have just described applies to any G-invariant Hamiltonian. A SIMPLE EXAMPLE: GEODESIC MOTIONS ON THE SPHERE 74 A simple example: Geodesic motions on the sphere We now describe a simple example for which one can give an explicit realization of the sym plectic cross section. Recall (see, e.g., Chap. 3 of Part 1) that the phase space for geodesic motions on 3 2 can be identified with a Hamiltonian G-space (T3 2 ,w,S0(3),J) where J: T3 2 -+ .so(3) * ~ JR 3 is given by J(q, v) =q xv . (Here we are viewing T3 as {(q, v) 3 x JR I q • v = 0}.) 2 E 2 3 If we restrict attention to the open dense invariant subset P C T 5 2 obtained by removing the zero section, we obtain a Hamiltonian G-space (P, w, S0(3), J) with regular momenta. Choose T, W and O as in Example 8.6. Then the symplectic fibration rr : P-+ 3 2 is given by rr(q, v) =rro(J(q, v)) = v/ llvll- The symplectic cross section F = rr- 1 ( e3 ) is sym plectomorphic to the cylinder 3 1 x (0, oo), equipped with the standard symplectic structure d0 I\ dl. Indeed a diffeomorphism <p: 5 1 x (0, oo) -+Fis given by where R~ = exp(0e 3). To show that <pis symplectic, recall that w = -d0 where 8.13 From this one computes <p*w 10 J <p*0 = J and 11 J <p*0 = 0, so that <p*0 I d0, giving = d0 I\ dl as required. The action of T ~ 5 1 on 5 1 x (0, oo) that makes the symplectomorphism <p a Tequivariant map is given by 0' • (0, I) = (0 + 0', I). Identify the Lie algebra of 5 1 (and its dual) with JR using the generator f0 (0). Then the momentum map JF o <p of this action is given by the coordinate function I: 5 1 x (0, oo)-+ JR. To summarize, the symplectic cross section of the Hamiltonian G-space applying to geodesic motions on 5 2 is just the cylinder 3 1 x (0, oo) equipped with the standard 5 1 Hamiltonian action. The Hamiltonian for geodesic motions on 5 2 is (up to a rescaling) H(q, v) = ½llvll 2 , which is SO (3)-invariant. This Hamiltonian restricts to a function on F that is pulled REDUCTION TO THE ABELIAN CASE back by cp : 5 x (0, oo) ~ F to ½1 1 75 This is the Hamiltonian for geodesic motions on 2 . the circle, and is 5 1-invariant. Thus geodesic motions on the sphere can be imagined as a family of subsystems F11 (v E 0), each identifiable with geodesic motions on the circle. This corresponds precisely to the following familiar fact: A point mass constrained to move on the surface of a smooth sphere moves as if it were constrained to move on a great circle. This circle is given by the intersection of the sphere with the plane whose normal is aligned with the initial (and subsequently conserved) angular momentum. These normals live in the space 5 2 = 0, the 'unit momentum sphere.' Reduction to the Abelian case Here is the central result of this chapter. 8.14 Theorem Two Hamiltonian G-spaces (Pi,w ,G,J 1 ) 1 and (P ,w ,G,J 2 2 ) 2 with regular momenta are equivalent (in the sense of 8.1) if and only if their symplectic cross sections are equivalent. Explicitly, if cp : F -+ F is an equivalence between the symplectic cross sections, then a 1 2 well-defined equivalence <p : Pi -+ P2 is given by </>(g. y) PROOF. =g. cp(y) (g E G, y E Fi) Recall that as G acts transitively on the fibers of 11' 1 : P 1 -+ 0 (the symplectic fibration associated with (Pi,w 1,G,J 1)), ever:y element of Pi is of the form g • y for some g E G and y E Fi = 11'11 (µo). Furthermore, g • y = g' • y' (g' E G, y' E Fi) if and only if g' = gq and y' = q- 1 • y, for some q E T (by the G-equivariance of 11' 1 ). In that case g' • cp(y') = gq • cp(q- 1 • y) = g • cp(y), by the T-equivariance of cp. This shows that ¢ is well-defined. That ¢ is smooth is clear. It is a diffeomorphism since it has a well-defined and smooth inverse ¢- 1 : P 2 -+ P 1 given by By construction ¢ is G-equivariant. THE 1r-INVARIANT T-ACTION 76 We show next that J 2 o </> = J 1 . Let g • y E P 1 be arbitrary (g E G, y E Fi). Then J 1 (y) E WC i, so that J 1 (y) = i- 1 (J 1 (y)lt) = i- 1 (JF1 (y)). Similarly, as <p(y) E F 2, we have J 2(1.p(y)) = i- 1 (JF2 (<p(y))). But JF2 01.p = JF1, since <p: F 1 ➔ F2 is an equivalence, from which we deduce 8.15 Using this fact we now compute Since g • y E F 1 was arbitrary, this shows that J 2 o </> = J 1 . To prove that</> is an equivalence it remains to show that</> is symplectic. Since <p : F 1 ➔ F 2 is symplectic, it follows from the definition of</>, and the last statement in Theorem 8.8, that</> maps fibers of 1r 1 symplectomorphically onto fibers of 1r 2. To check the symplecticity of </>, it therefore suffices to check that the tangent map T</> maps horizontal spaces of the symplectic connection on 1r 1 : P 1 ➔ 0 symplectically onto horizontal spaces of the symplectic connection on 1r2 : P2 ➔ 0. From the definition of</> and infinitesimal generators, it follows that (~ E fl, g E G, y E F) . 8.16 Since J 2 (</>(g · y)) = J 1 (g • y) (proven above), the symplecticity of T</> on horizontal spaces follows immediately from 8.16 and Lemma 8.12. □ The 1r-invariant T-action The remaining results of this chapter play a minor role in the sequel and may be skipped on a first reading. We call an action of a group H on P 1r-invariant if the fibers of 1r are H-invariant submanifolds. The action of T (as a subgroup of G) is not 1r-invariant, but it does leave the symplectic cross section F invariant. We now show how to extend this action on F to a 1r-invariant action. It is convenient to begin with a slightly more general result: THE n-INVARIANT T-ACTION 77 8.17 Lemma (Extension Lemma) Let (P, w, G, J) be a Hamiltonian G-space with reg- ular momenta, 1r: P---+ 0 the associated symplectic fibration, and (F, wF, T, JF) the symplectic cross section. Suppose that an Abelian group H acts on Fin a Hamiltonian fashion and let K : F ---+ Q* be an associated equivariant momentum map. Assume furthermore that: l. The actions of H and T commute, and 2. K is T-invariant. 3. JIF is H-invariant. Then, denoting the action of H on F by (h, y) f----t h·HY, there exists a well-defined extension to P given by (h E H, g E G, y E F) . The extended action of H leaves the fibers of 1r invariant and is Hamiltonian . An associated momentum map is the well-defined extension of K to a map K: P---+ Q* given by K(g • y) =K(y) (g E G, y E F) Moreover, the properties 1 and 2 above extend as follows: 4. The extended action of H commutes with the action of G , and 5. The extended momentum map K : P ---+ Q* is G-invariant. 6. J is invariant with respect to the extended action of H. To prove the lemma (see Appendix C) one appeals to Theorem 8.8 and Lemma 8.12. Taking H =T, we obtain an extension of the action of Ton F to a 1r-invariant action on P, with momentum map K: P---+ t* given by K(g. y) = J F(y) = J(y)lt= i(J(y)) = i(1rw(g • J(y))) = (i o 1rw o J) (g • y) = j(g. y) by 8.5 by the equivariance of J (j is defined in 8.7) . THE 1r-INVARIANT T-ACTION 78 Notice also that we can write q·r(g • y) = gqg- 1 - (g •y) (q ET, g E G, y E F). Summarizing: 8.18 Corollary If (P, w, G, J) is a Hamiltonian G-space with regular momenta, and 1r the associated symplectic fibration, then there exists a well-defined Jr-invariant action (q, x) f--7 qrx of Ton P given by (q E T, g E G, y E F) . This action is Hamiltonian with momentum map j : P--+ ~ C t * defined by j = i o 1rw o J . The actions of T and Gv on a fiber Fv (v E 0) are conjugate in the following sense: Let g E G be such that v = g • µo and define the isomorphism 'I/;: T--+ Gv by '1/;(q) = gqg- 1 . (Recall that T = Gµ 0 , so that gTg- 1 = G 9 .µ 0 = Gv.) Then qTx = '1/;(q) • x for any x E Fv. CHAPTER 9 Action-group coordinates as a symplectic cross section One can use symplectic cross sections as a way of generating new symplectic manifolds. In this chapter we show that the action-group model space G x ~ described in Part 1 can be realized as the symplectic cross section of T*G (with its 'irregular' points removed). We study the space G x ~ in some detail, supplying proofs of some facts stated in Part 1. The symplectic cross section of T*G Let 8 denote the canonical one-form on T*G. This is defined by 9.1 (( E T(T*G)) . The maps TT•G : T(T*G) -+ T*G and Ta : T*G-+ G denote the canonical projections. A natural symplectic structure on T*G is w = -d8. A (left) action of G on itself is given by (g, h E G) , where R 9 (h) = hg. Recall that the (covariant) cotangent lift T*¢ T*G -+ T*G of a diffeomorphism ¢ : G -+ G is defined by We make G act on T*G by cotangent-lifting the above action of G on itself: (g E G, x E T*G) . It follows from general results concerning cotangent-lifted actions (see, e.g., Marsden and Ratiu (1994)) that the above action of G on T*G is Hamiltonian, with an equivariant 79 THE SYMPLECTIC CROSS SECTION OF T*G 80 momentum map J : T*G-+ g* given by (xET*G,g=rc;(x)), where L 9 (h) =gh (g , h G). The open dense G-invariant subset P =J- (g;eg) T *G is a 1 E C Hamiltonian G-space with regular momenta. Each element of P is of the form T *Lg • µ for some g E G andµ E g;eg· (Indeed (g , µ) M T *L 9 • µ: G x g;eg -+ Pis a diffeomorphism.) The symplectic fibration 1r : P-+ 0 is given by Therefore, the symplectic cross section of (P, w, G, J) is given by For each a E g*, let aa denote the left-invariant one-form on G with aa(ida) = a (here viewing one-forms on G as sections of G -+ T*G). Then the embedding ¢ : G x to y T*G defined by ¢(g,p) = (i- (p))a(g) = T *L i- (p) is a diffeomorphism onto F. (Since 1 9 • 1 Theorem 8.8 says that F C P is a symplectic submanifold, this proves Proposition 2.2 stated in Part 1.) We define We =¢*w' and call (G to' we) the action-group model space X of G. In deriving an explicit formula for the symplectic structure We, it will be convenient to have a concrete way of expressing vectors tangent to G x t~. To this end, define for each (t, r) E g X t* the vector field (t, r)vf on G X to by 9.2 (t, r)vr(g,p) =ddt (gexp(tt),p+ tr)I t= O Since((, r) M (t, r)vr(g,p): g x t* -+ T( g,p)(Gxto) is an isomorphism at every (g,p) E Gxto, every vector tangent to G X to is uniquely expressible in the for~ (t , r)vr(g,p). One verifies that 1 9.3 1 We identify the Lie algebra of a Lie group with the left invariant vector fields, equipped with Jacobi-Lie bracket; our sign convention for the latter is the same as Abraham, Marsden and Ratiu (1988, §4.2.20). THE SYMPLECTIC CROSS SECTION OF T*G 81 9.4 Lemma Define Ga= </>*8, so that we= -dGa, Then (Ga,(~, r)vc(g,p)) = (p, a(~)) , where a : g-+ tis the projection onto t along t.L = [g, t]. Furthermore, The above expression for Ga also appears in Dazord and Delzant (1987, Section 5) (who obtain it via a different route). PROOF. One computes using 9.1 (Ga,(~, r)vc(g,p)) = (8, T</> • (~, r)vc(g,p)) = \ T *L 9 • i- 1 (p), :t Tc; ( </>(g exp(t~), p +tr)) lt=O) = (i- 1(p),TLg-l' :tTc;(T*Lgexp(te) -i - 1 (p+tr))lt=J = \i-l(p), :tg-1gexp(t~)1t=J = (i-l(p),O = (p,a(()) . 9.5 From this we compute Applying the well-known identity VJ u J d/3 = u J d( VJ /3) - V J d( u J /3) - [u, V] J /3 ' with /3 = Ga, we obtain from 9.3, 9.5 and 9.6 the formula for we stated in the lemma. □ 9. 7 Remark If G is a torus 1'k, then G x fa ~ ']["k x IR k and We is identifiable with the canonical symplectic structure I:j dqj I\ dpj. In this sense G X -lo (Gan arbitrary compact connected Lie group) is a non-Abelian generalization of action-angle coordinates. HAMILTONIAN VECTOR FIELDS IN ACTION-GROUP COORDINATES 82 Action-group coordinates as a Hamiltonian G-space Since we have realized G X fa as a symplectic cross section, the maximal torus T acts on G x to- More significant for us, however, is the observation that G acts on G x to: 9.8 Lemma The action ofG on G X to defined by g • (h,p) = (gh,p) is Hamiltonian, with equivariant momentum map JG: G x to-+ g* given by JG(g,p) = g-i- 1 (p) = Ad;_ 1 i- 1(p). This action makes (G X u, We, G, JG) a Hamiltonian G-space with regular momenta for any open set U C toPROOF. Imitate the construction in Example 3.2 of Part l. □ 9.9 Remark The action of Ton the Hamiltonian G-space G x to given by Corollary 8.18 is defined by q·r (g,p) = (gq,p). Corollary 8.18 states that this action is Hamiltonian , with a momentum map jG: G x to-+ to given by jG(g,p) = (i o 1rw o JG)(g,p) = p. The following proposition, whose easy proof is left to the reader, shows that the symplectic cross section of the action-group model space of G is the action-group model space of T, i.e., conventional action-angle coordinates (by Remark 9.7). 9.10 Proposition The symplectic cross section of (G X u, We, G, JG) is (T X u, Wy, T, JT). Hamiltonian vector fields in action-group coordinates In our applications to perturbation theory described in Part 1, it was essential to have a concrete way of writing down the equations of motion in action-group coordinates. These equations were derived from an expression for Hamiltonian vector fields that was stated without proof. We now derive that expression explicitly. In Part 1 it was convenient to identify G x to with G x to using some Ad-invariant inner product on g. Here we work throughout with G x fa and leave it to the interested reader to translate our results into the form used in Part 1. HAMILTONIAN VECTOR FIELDS IN ACTION-GROUP COORDINATES 9.11 Lemma Write t.L 83 =[g, t] and i.L =Ann t (notation as in Chap. 1 of Part 1). Then for all p E t0 the map is an isomorphism. PROOF. Since dim t.L = dim i.L' it suffices to show that ade i- 1 (p) = 0 implies ( = 0. Write µ = i- 1 (p) E W C in g~eg, so that t = gµ (by, e.g., Corollary 1.8.2 of Part 1). Supposing that ( E t.L satisfies ade i- 1 (p) = 0, we have ade µ = 0, i.e., ( E gµ = t. But tn t.L = {0} (by, e.g., Theorem 1.2.8 of Part 1), so that ( = 0. For any p E □ ta, we define Ap : i.L -+ t.L to be the inverse of the map described in 9.11 above. For future reference, let us record that this means 9.12 For any smooth function f : G x t0 -+ IR. define the vector-valued functions g* and i: :G x t0-+ i~ :G x ta -+ t by d ( &J &g (g, p), O = (df, ((, 0)vr(g, p)) = d/(g exp(t(), p) I t=O d (r, &J &p (g, p)) = (df, (0, r)vr(g, p)) = d/(g, P + tr) t=O I (T 9.13 Proposition (Hamiltonian vector fields on G x t0) The Hamiltonian vector field X J corresponding to a function f is given by where E t*) . THE POISSON BRACKET IN ACTION-GROUP COORDINATES 84 Here a* : fl* ---+!denotes the projection along !1_ = Ann t, i.e., (a*µJ,) = (µ, at) (µ E fl*, t E fl)· PROOF. Define t E fl and TE t* as in the statement of the lemma. It suffices to verify that the one-forms (t, r)vf J Wa and df agree on elements of T(g,p)(G X fa). Let X E fl and y E t* be given. Then, by Lemma 9.4, ((t, r)vf J Wa, (x, Y)vr(g, p)) = \Y, at) - (r, ax)+ \P, a[t, x]) 1 = (ia*!; (g,p), ax)+ (y, !; (g,p)) + (i- (p), [Ap !; (g,p), x] + [!; (g,p), x]) 9.14 = ( 8J (g,p),ax)+ (y, 8J ·-1 8J Bp(g,p))+ (i (p),Ap (g,p),x]) 89 89 We have used the fact (i- 1 (p),[~t(g,p),x]) = 0, which is true because i- 1 (p) E Ann[t,fl] and ~t(g,p) Et. Noting that Ap~t(g,p) E t1- and using 9.12, (i- 1 (p), [Ap ~; (g, p), x]) = (>.; 1 Ap ~; (g, p), x) = ((id - a*)!; (g,p), x) = ,!; (g,p), (id - a)x) , so that 9.14 becomes * 8J 8J ((t, r)vf J wa, (x, Y)vr(g,p)) = (Bg (g,p), x) + (y, Bp (g, P)) = (df, (x,y)vr(g,p)) Since x E fl and y E t* were arbitrary, this proves our assertion. □ The Poisson bracket in action-group coordinates The Poisson bracket on G x fa is defined by {J, h} = xh J X1 J Wa· 9.15 Corollary (The Poisson bracket on G X fa) Dropping the '(g, p) 'argument from '~t (g, p),' etc., the Poisson bracket on G x fa is given THE POISSON BRACKET IN ACTION-GROUP COORDINATES 85 by where Ap : g* ➔ g is the map defined in Lemma 9.13 above. PROOF. Imitate the proof of Lemma 2.15 of Part l. □ CHAPTER 10 Constructing action-group coordinates Noticing that equivalent Hamiltonian G-spaces (see 8.1) have identical momentum map images, we introduce the following terminology : 10.1 Definition Let (P, w, G, J) be a Hamiltonian G-space and define V = J (P) and U = i(V n W) . (If G is Abelian, then U = V .) We say that P admits G-compatible action-group coordinates if (P,w , G , J) is equivalent to (G x U,wc , G,J 0 ) . We sometimes distinguish the Abelian case (G, a torus) by saying that P admits G-compatible action-angle coordinates. The non-Abelian case Here is our main result. It follows immediately from Theorem 8.14 and Proposition 9.10. 10.2 Theorem A Hamiltonian G-space (P, w, G, J) with regular momenta admits G-compatible actiongroup coordinates<=> its symplectic cross section (F, wp, T, JF) admits T-compatible actionangle coordinates. Indeed if U denotes the image of JF , and <.p: T x U ~Fan equivalence, then a well-defined equivalence ¢ : G x U ~ P is given by (g E G, y E F) The existence of action-angle coordinates can be characterized as follows. 10.3 Proposition Let T be a torus and (F, w , T , J) a Hamiltonian T-space. Then F admits T-compatible action-angle coordinates<=> the following conditions hold: 1. T acts freely and the space is geometrically integrable (see below). 86 THE NON-ABELIAN CASE 2. J : F-+ U 87 =J(F) has connected fibers. 3. J : F -+ U admits a (global) isotropic section s : U -+ F. Recall that a section s : U-+ F is isotropic if s*w = 0. As in Part 1, when a group G acts freely on a Hamiltonian G-space (P,w,G, J) , we call the space geometrically integrable if the Marsden-Weinstein reduced spaces are zero dimensional. If J(P) C fl;eg, then geometric integrability occurs precisely when dim P = dim G + rank G (see Chap. 3 of Part 1 for details). In particular, in Proposition 10.3 (where G = T) this means dim F = 2 dim U. In that case the image of a section s : U -+ F has half the dimension of the ambient space. A section whose image has this dimension and is isotropic is usually called a Lagrangian section. PROOF OF ⇒. That conditions 1 and 2 hold is obvious. Condition 3 follows since the map pi----+ (q,p): U-+ T x U is an isotropic section (for any q ET). □ PROOF OF ¢:. Since Tacts freely, J : F-+ U is a surjective submersion (apply identity 8.3). By the pre-image theorem the fibres of J are submanifolds of F of dimension dim F dim T. Since the space is geometrically integrable this dimension is dim T. By momentum map equivariance - which amounts to invariance since T is Abelian - the connected components of a fibre are therefore T-orbits. It follows from condition 2 that each fibre is a single T-orbit. Consequently, as T acts freely, the map ¢ : T x U -+ F defined by q;(q, p) =q • s(p) is a diffeomorphism. Now¢ is clearly T-equivariant, and we have (Jo q;)(q,p) = p = JT(q,p), since J is T-invariant and s is a section. So to show that¢ is an equivalence, it remains to show that ¢ is symplectic. This means showing that <p*w = wT where, according to Lemma 9.4, wT is given by 10.4 ( (q ,p) ET XU; 6,6 Et; T1,T2 Et*) . THE NON-ABELIAN CASE 88 We will compare </>*wand wy on pairs of vectors of the form((, T)vr(q,p). We start with a simple computation: d T</> • ((, T)vr(q, p) = dt </>(qexp(t(), p + tT) lt=O d = dt exp(t() • (q • s(p + tT)) l t =O (Tis Abelian) d =(F(</>(q,p))+(T<I>qoTs)· dt(p+tT)lt=O, where the diffeomorphism <I> q : F ➔ F is defined by <I> q ( x) = q • x . Using this we compute 10.5 </>*w( (6, T1)vr(q,p), (6, T2)vr(q,p)) = w( (6)F(</>(q,p)), (6)F(</>(q,p))) + w ( (6)F(</>(q,p)),(T<I>qoTs) • :/p+tT2)lt=O) -w ( (6)F(</>(q,p)), (T<I>qoTs) • :/p+tTi)lt=O) + w ( (T<I>q o Ts) • :t (p + tTi) lt=O' (T<I>q o Ts)· :/P + tT2) l t=O) The first term on the right-hand side vanishes by 8.12.1 , since Tis Abelian. The last term vanishes because <l>q is symplectic and s : U ➔ F is isotropic. The remaining terms are of the form w ( (F(</>(q,p)), (T<I>qoTs) • :t(p+tT)lt=O) = w ( (F(q,p), Ts· :t (p + ts) it=O) since <I>q is symplectic and <I>;(F = (F d = (dJt, Ts· dt (p d + tT)) since (F = XJe = (dt(J(s(p + tT)),() = (T,() since Jo s = id (sis a section). From this we see that 10.5 and 10.4 have the same right hand sides, so that </>*w = wy as claimed. □ Of course our main interest in Proposition 10.3 is the case in which Fis the symplectic cross section of some Hamiltonian G-space P (G non-Abelian). In that case, some of the THE NON-ABELIAN CASE 89 conditions listed in 10.3 can be cast as conditions on the space P. This will be useful later on. 10.6 Lemma Let (P, w, G, J) be a Hamiltonian G-space with regular momenta and (F, WF, T, JF) its symplectic cross section. Then 1. T acts freely and (F, WF, T, JF) is geometrically integrable {:} G acts freely and (P, w, G, J) is geometrically integrable. 2. Every fiber of JF: F-+ U =JF(F) is homeomorphic to fiber of J: P-+ V =J(P) a and conversely. (In particular, JF has connected fibers {:} J has connected fibers.) PROOF. That TC G acts freely on F C P when G acts freely on Pis trivial. Conversely, suppose T acts freely on F, and let g E G be such that g • x = x for some x E P. Then by the equivariance of the symplectic fibration rr : P -+ 0, we have g E Gv, where v = rr(x). Since G acts transitively on 0, there is h E G such that v = h • µ 0 . Then Gv = Gh•µ,o = hGµ, 0 h- 1 = hTh- 1 , so that g = hqh- 1 for some q ET. Then g ·x = x implies q • (h- 1 • x) = h- 1 • x. Since h- 1 • x E Fµ, 0 = F (again by the equivariance of rr), and we are supposing that T acts freely on F, it follows that q = id, so that g = hh- 1 = id. This shows that G acts freely. Since dim P = dim O + dim F = (dim G - rank G) + dim F, and rank G = dim T, we have dim F = 2 dim T {:} dim P = dim G + rank G . This shows that F is geometrically integrable if and only if P is geometrically integrable. So 1 is true. Each fiber of J lies entirely in a fiber of the symplectic fihration rr : P -+ 0, and is therefore (by, e.g., Theorem 8.8) homeomorphic to a fiber of J lying entirely in the fiber F = Fµ, 0 • Now by definition JF = i(JIF), where i : i-+ t* is an isomorphism. So every fiber of J F is literally a fiber of J, and every fiber of J is homeomorphic to a fiber of J F. So 2 is true. □ THE NON-ABELIA N CASE 90 The following observation is useful in constructing explicit action-group coordinates in concrete examples: 10. 7 Proposition Let (P, w, G, J) be a Hamiltonian G-space with regular momenta and (F, wp, T, JF) its symplectic cross section. Assume that F admits T-compatible actionangle coordinates (so that P admits G-compatible action-group coordinates, by Theorem 10.2) and let s : U---+ F be a Lagrangian section (whose existence is guaranteed by Proposition 10.3). Then a realization of G-compatible action-group coordinates is given by the equivalence</>: G x U ~ P defined explicitly by </>(g,p) =g • s(p). PROOF. Let s : U ---+ F be a Lagrangian section. As in the proof of Proposition 10.3, we have an equivalence cp : T x U ~ F given by cp(q,p) =q • s(p). Applying Theorem 10.2, a well-defined equivalence </> : G x U ---+ P is given by </>(g. (q,p)) = g. cp(q,p) (g E G, q E T C G, p E U) But this means </>(gq,p) = (gq) • s(p) for any g E G, q ET, p EU. In particular </>(g,p) = □ g • s(p), as claimed. 10.8 Example (Geodesic motions on 5 2 ) In the example of geodesic motions on 5 2 discussed in Chap. 8, the map cp : 5 1 x (0, oo) ---+ F defines an equivalence between the space (TX U, WT, T, JT) = (5 1 x (0, oo), d0 A dl, 51, I) and the cross section (F,wp,T,JF). In other words, F admits T-compatible action-angle coordinates. The map I 1----t ( 0,I) : (0, oo) ---+ S 1 x (0, oo) is a Lagrangian section for any 0, so that a Lagrangian section s : (0, oo) ---+ F is given by The action-group model space for G is (up to the obvious identifications) S0(3) x (0, oo) (see, e.g., Example 2.14, Part 1). Applying Proposition 10.7, a realization of G-compatible THE ABELIAN CASE 91 action-group coordinates for P is given by the equivalence ¢ : SO(3) x (0, oo) ➔ P defined by The SO(3)-invariant Hamiltonian H(q, v) = ½llvll is pulled back to Ho </>(g,p) = h(p) 2 = ½p 2 . (That this problem admits action-group coordinates was observed more or less directly in Remark 3.5, Part 1.) The Abelian case Theorem 10.2 reduces the construction of action-group coordinates to that of actionangle coordinates. The remainder of this chapter revisits in some detail the classical problem of constructing action-angle coordinates, from the present Hamiltonian G-space point of view. (The necessary and sufficient conditions given in Proposition 10.3 are not always very practical.) Suppose then that (F, w, T, J) is a Hamiltonian T-space on which a torus T is acting freely, and assume that the space is geometrically integrable. These assumptions are easy to check. Then, as we observed in the proof of Proposition 10.3, J : F ➔ U =J(F) is a surjective submersion, and the connected components of the fibers of J are T-orbits. Now unless one assumes that the fibers are compact, serious technical difficulties impede further progress 1 . Similar difficulties arise in attempts to generalize momentum map 'convexity theorems' to the non-compact case (see, e.g. , Hilgert et al. (1994) 2). If the fibers of J are compact, or equivalently, have a finite number of connected components 3 , then by a corollary of the Reeb stability theorem (see, e.g., Camacho and Neto (1985)), J: F ➔ U is a locally trivial fiber bundle. We consider two scenarios: 1 2 Unless one works 'locally'. See, e.g., Remark 3.11, Part l. Note that these results do not apply immediately here since we cannot assume that J is proper as a map into t•. 3 Remember that these fibers are regular submanifolds, by the preimage theorem. THE ABELIAN CASE 92 Scenario 1. Suppose that J : F -+ U has connected components. (Checking this is not always trivial.) Then J : F -+ U is a Lagrangian fibration with connected fibers, to which the study of Duistermaat (1980) is applicable. Recall that a submanifold L C F is Lagrangian if the inclusion l : L Y F is isotropic (l*w = 0) and if dim F = 2 dim L. That the fibers of J are Lagrangian follows from 8.12.2 and our integrability assumption. Duistermaat characterizes the obstruction to the existence of a (not necessarily Lagrangian) section s : U -+ F in terms of an algebraic invariant v of the bundle known as a Chern class. A section exists if and only if v = 0. In that case the de Rham cohomology class [s*w] E H 2 (U) is the same for all sections s, and a Lagrangian section exists if and only if this class vanishes. A sufficient condition is that w is exact. See op. cit. for details. Scenario 2. In this case we weaken the connectedness hypothesis but enforce a topological assumption on the momentum map image. We formulate our conclusions in the form of a theorem, whose proof is constructive, up to the existence of a Riemannian metric. 10.9 Theorem Let (F, w, T, J) be a Hamiltonian T-space on which a torus T is acting freely, and assume this space is geometrically integrable (i.e., dim F = 2 dim T ). Assume F is connected and that each fiber of J : F -+ U = J ( F) is compact, or has a finite number of connected components. Assume that U is smoothly contractible. Then each fiber is in fact connected, and there exists a Lagrangian section s : U-+ F of J : F-+ U. In particular, F admits T-compatible action-angle coordinates. Note that in examples it is often easy to check the compactness of the fibers of J. The Euler-Poinsot rigid body (see Chap. 11) is a case in point. PROOF. The last statement follows from the preceding ones and Proposition 10.3. We begin by showing that the fibers of J are connected. By the ·hypotheses and our earlier remarks, J : F -+ U is a locally trivial fiber bundle, each fiber being a finite number of T-orbits. Proceed by constructing a covering space j : U -+ U as follows: For each p E U let Qp denote the set of connected components of J- 1 (p), and for each x E F let [x] denote the element of QJ(x) that contains x. Define U = UpEUQp and j : U -+ U by THE ABELIAN CASE 93 J([x]) = J(x) . Using the local triviality of J: F ➔ U and the connectedness of F, it is not difficult to show that U admits the structure of a smooth connected manifold with respect to which j : U ➔ U is a (smooth) covering map. The multiplicity (number of sheets) of this cover is the number of connected components in each fibre J- 1 (p). But since Uc to is contractible, it is simply connected . From covering space theory it follows that j : f) ➔ U is a diffeomorphism . The multiplicity of the cover is therefore one, proving that the fibres of J are connected . The next step is to construct a section for J : F ➔ U. Equip the bundle J : F ➔ U with an Ehresmann connection (see, e.g., Kobayashi and Nomizu (1963)). This is always possible by giving F the structure of a (smooth) Riemannian manifold. That one can do so in the C 00 category is a standard partition of unity argument. In the real-analytic category this follows from the Whitney-Morrey-Grauert Embedding Theorem (Grauert, 1952) . Now U contracts to some point Po E U. That is, we have a smooth map (p, t) f---t Kt(P) : U x [O, 1] ➔ U with Ko= Cp 0 and K1 = idu, where Cp 0 : U ➔ U is the constant map Cp 0 (p) =Po - Fix any xo E J- 1 (p 0) and define a smooth section s : U ➔ Fas follows: For any point p E U define a path 1 : [O, 1] ➔ U from Po top by ,(t) = Kt(P)- Use the connection to lift I to a path i: [O, 1] ➔ F with i(O) = xo, and define s(p) = -y(l). Since the fibers of J are precisely the T-orbits, and T acts freely, the smooth map </>: T x U ➔ F defined by </>(q,p) =q • s(p) is a diffeomorphism. Our final step will be to exhibit a Lagrangian section s : U ➔ F. This section will be constructed by modifying the existing section s using a 'Lie transform' argument appearing in Guillemin and Sternberg (1984, §44). This argument is in the spirit of proofs and generalizations of Darboux's theorem given by Moser (1965) and Weinstein (1971). To this end, we first show that w is exact. Indeed, as J : F ➔ U is globally trivial and U is contractible, there exists a smooth deformation retract of F onto some fiber T'. The fibres are just T-orbits, which are Lagrangian since T is Abelian (apply the formula in 8.12.2). Because of the deformation retraction, the inclusion i : T' ➔ F induces an isomorphism in THE ABELIAN CASE 94 de Rham cohomology. Since T' is Lagrangian l*w = 0, implying that the cohomology class of w vanishes, i.e., w = d0 for some one-form 0. We are now ready to construct the Lagrangian section. Suppose ct is a one-form on U. Then at each point p E U C t* one may associate to ct an element &(p) E t defined through (v, &(p)) = (ct, v) (v E T pU ~ t*) . One may next associate to ct a vector field X°' on F, defined by X°'(x) = (&(p))F(x) ( X E F, p = J (x) ) , where '/]F denotes the infinitesimal generator of the T action associated with an element 'I] E t. In particular, by the definition of a momentum map 10.10 (X°' J w, v) = (dl&(p), v) The vector field X°' is tangent to the T-orbits which, as shown above, are precisely the fibers of J. These fibers are compact, so that we have a well-defined flow associated with X°', which we denote by (x,t) t-+ ctt(x): F x (-00,00)--+ F. The time-one map ct 1 is a diffeomorphism of F preserving the fibers of J : F --+ U. In general ct 1 is not sym plectic. The strategy is to defines= ct1 os and determine how ct might be chosen such that s*w = 0. We have = s*w + s* fa\ :t ct;w )dt 1 = s*w + s* fo (ct;(X°' J dw) + ct;d(X°' J w))dt 1 = s*w + fo s*ct;d(X°' J w)dt 1 10.11 = s*w + fo d(s*ct;(X°' J w))dt THE ABELIAN CASE 95 For any u E TpU ~ t* we have = (dJ&(p), T(at o s) • u) = .!!:....1 &(p) ( ( lYt dT O by 10.10 s) (p + TU)) I r=O = ddT ((Jo lYt o s)(p + Tu) , &(p))lr=O = (a, :T (p + Tu) lr=O) = (a, u) . Therefore s*a;(xa J w) = a. Using this in 10.11 gives s*w = s*w + da. Choosing a= s*8 gives s*w = 0, so that sis a Lagrangian section. □ The following corollary was stated without proof as Theorem 3.10 in Part 1. The only difference here is that we have not made the identification G x t~ ~ G x to that was convenient in our applications to perturbation theory. Recall that i : i -=:'...+ t* is the restriction to i = Ann[_g, t] of the natural projection _g* --+ t*. 10.12 Corollary Let (P, w, G, J) be a Hamiltonian G-space on which G is acting freely, and for which J(P) C fl;eg . Assume the space is geometrically integrable (i.e., dim P = dimG + rankG), that Pis connected, and that each fiber of J: P--+ J(P) is compact or has a finite number of connected components. Assume that U = i(J(P) n W) (which is open in~) is smoothly contractible. Then the spaces (P,w,G,J) and (G x U,w'c;,G,JG) are equivalent, i.e., P admits G-compatible action-group coordinates. 10.13 Remark Since J(P) C fl;eg is G-invariant, i(J(P) n W) = j(P), where j: P--+ t* is the map defined by 8.7. This fact is useful in computations. PROOF. Let (F,wF,T,JF) denote the symplectic cross section. Since G acts freely on P and (P, w, G, J) is geometrically integrable , T acts freely on F and (F, WF, T, J F) is geometrically integrable, by Lemma 10.6.1. By the definition of JF , we have i(J(P) n W) = JF(F), so that U = JF(F) is contractible by hypothesis. By Lemma 10.6.2 and the THE ABELIAN CASE 96 hypothesis on the fibers of J, each fiber of JF is compact or has a fin ite number of connected components. We have therefore established that the hypotheses of Theorem 10.9 apply to the symplectic cross section, which therefore admits T-compatible action-angle coordinates. Theorem 10.2 finishes off the proof. D CHAPTER 11 The axisyrnrnetric Euler-Poinsot rigid body In this chapter we apply the general results of the preceding chapter to a nontrivial example, the Euler-Poinsot rigid body. Problem prescription The Euler-Poinsot rigid body is a rigid body fixed at but free to rotate about a point 0 that is motionless in some inertial frame of reference. It is well-known that the dynamics of such a body is described by integral curves of an appropriate Hamiltonian vector field on T* S0(3). We refer the reader to, e.g., Marsden (1992, p. 87) for details . Note that one ordinarily identifies the phase space T* S0(3) with T S0(3), using an appropriate invariant metric (see below). Let A, p: S0(3) x JR 3 ~ T S0(3) denote the usual left and right trivializations: d tml t=O , A(A, m) -= dtAe d tn A It=O • p(A, n) -= dte Here~ t--+ f : JR 3 -+ .so(3) is the isomorphism defined by fu = ~ x u (u E JR 3 ). We think of tangent vectors as equivalence classes of curves and use ' ftAtlt=o' to denote the class represented by t t--+ At. In what follows, the vector m will correspond to the body angular momentum of the body abo ut 0, while n will correspond to the spatial angular momentum about 0. Note that 11.1 .X(A, m) = p(A, n) <=> n = Am 97 PROBLEM PRESCRIPTION 98 Define a one-form 0 on T SO(3) as follows. An arbitrary vector tangent to T SO(3) is of the form -9t p(At, nt) lt=O· Let n' be the element of JR 3 uniquely determined by d -dAt t I t =O I =p(Ao,n). Then by decree, 11.2 where the dot denotes the usual Euclidean dot product. An invariant Riemannian metric on SO(3) is given by The differential form 0 above is just the canonical one-form on T* SO(3) (see Chap. 9), viewed as a one-form on TSO(3) using the ((·,·))-induced identification T*SO(3) ~ TSO(3). In particular, w = -d0 is a symplectic two-form on TSO(3). Define H : T SO(3) --+ JR by Then: The dynamics of an Euler-Poinsot rigid body with moments of inertia Ii, h, h about the fixed point O is described by the Hamiltonian system (T SO(3), w, H) above. For a proof, see op. cit. Assume that the moment of inertia ellipsoid is axisymmetric, i.e., that two of the moments of inertia are equal: Ji = h = I, say. In that case the Hamiltonian is invariant with respect to the following action of G = SO(3) x S 1 on T SO(3): for (A, 0) E SO(3) x S1, define (A, 0) • ,\(A, m) = ,\(AAR~, R~m) , EXISTENCE 99 where R~ = exp(0e3) is the rotation about the e3-axis through angle 0. Alternatively, we may write this action as (A, 0) • p(A, n) = p(AAR~, An) . The S0(3) part of the symmetry group corresponds to the familiar rotational 'spatial' symmetry of the system, while the S 1 action corresponds to the axisymmetry of the inertia ellipsoid (i.e., a 'body' symmetry). The action is Hamiltonian with equivariant momentum map J : T S0(3) -t g* ~ R 3 x R given by where m3 = m • e3. Alternatively, These formulas are consistent with Noether's theorem (stating that J is constant on solution curves) and the familiar fact that in an axisymmetric rigid body the spatial angular momentum and the component of body angular momentum along the symmetry axis are conserved. Our objective is to construct G-compatible action-group coordinates in the space (T S0(3), w, G, J). Existence Before constructing action-group coordinates explicitly, we convince ourselves that such coordinates must exist. In the first place, we claim: J has compact fibers. Indeed, suppose that y =(µ,a) E J(P) C R 3 x JR. Then certainly 1 J- (y) C .,\ ( S0(3) X Bl[µll(O)) , where Br(O) C R 3 denotes the closed ball of radius r. Since .,\ is a diffeomorphism, this shows that J- 1 (y) is contained in a compact set. Since J- 1 (y) is closed in TS0(3) (by the continuity of J), J- 1 (y) itself must be compact. EXISTENCE So that G acts freely and the image of the momentum map is contained in g;eg = R 3\{0} x R, we restrict attention to the open dense G-invariant set PC TS0(3) defined by P = {.X(A, m) I A E S0(3) and m E R 3 and (m 1 -=JO or m2 -=JO)} . That is, we remove from phase space those points whose body angular momentum vector m lies on the e3-axis. For a maximal torus T C G we choose T = {(R~,0) E S0(3) X 5 1 I </> E [0 ,2n),0 E 5 1 } ~ 1' 2 . Then i = span{ e3} x JR.. For a Wey! chamber Win g* choose W ={(te3,r) E R x RI t E (O,oo),r ER}. For a regular co-adjoint orbit choose O and O is µ 0 =(e3,0). 3 =5 x {0}. The unique intersection point of W 2 The projection rra : R 3\{0} x R -+ 0 ~ 5 2 is given by rro(µ, a) = µ/ 11µ11- The projection rrw: R 3\{0} x R-+ W ~ (0, oo) x R is given by rrw(µ, a)= (11µ11, a). Now dim P = 6 = 4 + 2 = dim G + rank G. So (P, w, G, J) is geometrically integrable. Since J has compact fibers, we can apply Corollary 10.12, provided we can show that U = i(J(P)nW) is smoothly contractible. By Remark 10.13, U =j(P), wherej = iorrwoJ. With our identifications, i is just the identity, so that we obtain j(.X(A, m)) = (llmll, m3). It follows that (see Fig. 1). In particular, U is indeed contractible. By Corollary 10.12, G-compatible action-group coordinates exist. THE SYMPLECTIC CROSS SECTION 101 P2 PI FIGURE l. The set U =i(J(P) n W) = j(P) = JF(F) in the axisymmetric Euler-Poinsot rigid body. The symplectic cross section Since P admits G-compatible action-group coordinates, the symplectic cross section F admits T-compatible action-angle coordinates (Theorem 10.2). We shall construct actiongroup coordinates explicitly by applying Proposition 10.7. To this end, we now construct a concrete realization of F, its symplectic structure WF, and the momentum map JF: F-+ U. The symplectic fibration 1r : P-+ 0 ~ S2 (see Theorem 8.8) is given by rr(p(A, n)) = n/ llnll- So the symplectic cross section is F = rr- 1 (e3) = {p(A, n) I A E SO(3) and m E JR 3 and (A- 1 n · e 1 =JO or A- 1 n · e2 =J 0) and n/ llnll = e3} = {p(A,p1e3) I A E SO(3)\Z, P1 > 0} , where Notice that if one constructs Euler angles on SO (3), according to the appropriate sign convention, then Z C SO (3), which is just the disjoint union of two circles, corresponds to the coordinate singularities. In particular, if we write SO(3)' 1" 2 X (0, rr). =SO(3)\Z, then SO(3)' ~ EXPLICIT ACTION-GROUP COORDINATES 102 The map 1/i : S0(3)'x (0, oo) -+ F defined by 1/i(A, p 1) = p(A- 1 , p 1e3) is a diffeomorphism onto F. Let iF: F c.....+ P denote the inclusion. Then 1/i*wF = -d0', where 0' = 1/i*ij;,0. A vector tangent to S0(3)' x (O,oo) is of the form -ft(Aexp(tt),P1 + tT)lt=o· One computes using our earlier definition of 0 (equation 11.2), d , 11.3 (0', d/Aexp(tO,P1 + tT)lt=O) = 1 (0, :tp(exp(-tt)A- , (P1 + tT)e3) lt=O) = -p(e3 • ~) The map JF: F-+ JR 2 is given by JF(p(A, n)) = (n 3 , (A- 1n) · e3) for all p(A, n) E F. So (JF o 1/i)(A, p 1) = (p 1, (Ae3 • e3)p 1). As in the proof of 10.12, JF(F) = U (computed above). Let T ~ 1' 2 act on SO (3) x (0, oo) in the way that makes 1/i a 1' 2-equivariant map. (We will not need to compute this action explicitly.) Then 1ji establishes an equivalence between the space (S0(3)' x (O,oo),-d0',1'2,J'), and the symplectic cross section (F, WF, 1' 2 , JF), where 0' is the one-form computed in 11.3, and J' = JF o 1/i: S0(3)' x (0, oo)-+ U is given by J'(A,p1) = (P1, (Ae3 • e3)p1). Explicit action-group coordinates To construct action-group coordinates by applying 10.7, it remains to exhibit a Lagrangian section for J F. Defines: U-+ S0(3)' x (0, oo) by where R~ = exp(ae 1) is the rotation about the e 1-axis through angle a. By convention, arccos is to take values in (0, rr). EXPLICIT ACTION-GROUP COORDINATES 103 One verifies immediately that J' os = idu, i.e., sis a section of J': SO(3)' x (0, oo)-+ U . We claim that s is in fact Lagrangian. Indeed, for any r 1 , r 2 E JR we have (s*8',ddt(P1+tr1,P2 +tr2)I t =O ) = (8' ,ddt(R 1 (P2+tr2),P1+tT1)I ) arccos Pi +tr1 t=O = (8' , :t (R~rccos( p2 /pi) exp(tf(P1, P2 , T1, T2) i 1), Pl+ tri) lt=O) for some real-valued function f. , It is irrelevant what f actually is since, applying the formula 11.3 for 8', we obtain Since r 1 and r 2 were arbitrary, s*8' = 0, implying s* (-d8') = 0. Sos is indeed Lagrangian. Since 'I/; : SO(3)' x (0, oo) -+Fis an equivalence, 'I/; o s: U-+ Fis a Lagrangian section for JF : F-+ U . One computes We may now apply Proposition 10.7 as follows: Define 1>: G x U = 50(3) x 5 1 x U-+ PC T5O(3) by = .\(AR~arccos(pifp 1 )R~, mp1,P2,0) , where mp 1,p2,0 = (-Jpf - p~ sin 0, - ✓Pi - p~ cos 0, p1) Then ¢ is an equivalence realizing G-compatible action-group coordinates in the space P. The Hamiltonian cf>*H pulled back to G x U is given by 1>*H(g,p) = h(p), where _12 11 12 h(p1,P2) = P1 + (h 21 2 1)P2 . Notice that his convex (in the sense of Definition 5.3, Part 1) in the prolate (h < I) case. CHAPTER 12 Converting dynamic integrability to geometric integrability Let (P, w, G, J) be a Hamiltonian G-space with J (P) C g;eg, and H : P-+ JR a fixed Gin variant Hamiltonian. Assume that G acts freely. Then all the Marsden-Weinstein reduced spaces J- 1 (µ)/Gµ (µ E J(P)) have the same dimension, viz. d = dim P- rankG - dim G. Recall that we call the space geometrically integrable (or G-integrable) if d = 0. We call the space dynamically integrable (or (G, H)-integrable) if d = 2. These definitions are motivated in, e.g., Chap. 3 of Part l. We have already shown (Lemma 10.6) that P is geometrically integrable if and only if its symplectic cross section Fis geometrically integrable. An analogous statement holds for dynamic integrability, which one proves in the same way. To construct action-group coordinates (see, e.g ., Corollary 10.12) one requires geometric integrability. In examples, however, it can happen that the symmetry group that presents itself is only large enough to enforce dynamic integrability. The objective in this chapter is to show how to extend the action of G in the dynamically integrable case to a geometrically integrable Hamiltonian action of G x S 1 , with respect to which H is G x S 1 -invariant. (Of course we can always extend the symmetry group by a factor JR by considering the flow of XH, assuming that XH is complete. The point is that we seek an extension that is compact.) The extra S 1 action is to be built in some way from the reduced dynamics of H, and consequently depends on the particular H of interest (a different G-invariant Hamiltonian will not necessarily be G x S 1 -invariant) . We shall not attempt great generality but focus on the case where the reduced dynamics of H in each (two-dimensional) reduced space is composed of an equilibrium point surrounded by periodic trajectories. This is true for the asymmetric Euler-Poinsot rigid body, if we restrict attention to an appropriate open subset of the phase space. For simplicity, we illustrate our algorithm in the axisymmetric 104 SETUP 105 rigid body, viewed as a (S0(3), H)-integrable space (see Appendix D). It turns out that the extra 5 1 action (which our algorithm constructs ab initio) corresponds to the familiar 5 1 symmetry of the body, as exploited, for example, in the previous chapter. We shall make assumptions in the general theory that lead to the fewest technicalities, even if these are inconvenient to check in applications. Setup With Lemma 8.17 in hand one can easily reduce the above 5 1 extension problem to the Abelian case by passing to the symplectic cross section. We therefore begin by restricting attention to the case of a toral symmetry. In this chapter T denotes a torus, (F, w, T, J) a Hamiltonian T-space on which T is acting freely, and H : F -+ ~ a fixed T-invariant Hamiltonian. We assume that (F, w, T, J) is dynamically integrable (i.e., dim F = 2 dim T + 2). For simplicity, we assume that J : F -+ J (F) has the property that preimages of connected sets are connected. This is true, for example, if J : F -+ J (F) is a locally trivial fiber bundle with connected fibers. (Note that J is already a surjective submersion, on account of 8.3). Denote by p : F -+ F /T the canonical projection. Since we assume that T acts freely and properly, F /T admits a differential structure with respect to which p is a surjective submersion. Since Tis compact, p: F-+ F/T is a principal T-bundle (by a corollary of the Reeb stability theorem; see, e.g., Camacho and Neto (1985)). We equip F/T with the unique Poisson structure with respect to which p is a Poisson map. Since J is T-equivariant (by the definition of a Hamiltonian T-space) and T is Abelian, it follows that J is T-invariant and therefore factors through p. That is, there exists a (unique) map j : F /T -+ t* with the property that the following diagram commutes: THE S 1 EXTENSION 106 F F/T t* =lj Since the Hamiltonian H is T-invariant , it drops to a function on F /T that we shall denote by h. To formulate and prove our claims, it will be convenient to realize all the MarsdenWeinstein reduced spaces as the symplectic leaves of F/T. This is possible (since T is Abelian) with the help of the following fact. 12.1 Lemma Let tj denote the image of J. Then under the assumptions above, j : F /T--+ tj is a surjective submersion whose fibers are the symplectic leaves in F /T. That j : F /T --+ tj is a surjective submersion follows from the fact that J : F --+ tj is a surjective submersion. The other conclusion of the lemma follows from the Symplectic Leaf Correspondence Theorem for dual pairs. See Appendix E. Let Eµ = j- 1(µ) be a symplectic leaf in F/T (µ E tj). Then as j- 1 (µ) = p(J- 1 (µ)) = J- 1 (µ)/T = J- 1(µ)/Tµ, the leaf Eµ is literally a Marsden- Weinstein reduced space. Of course each leaf Eµ C F/T is Xh-invariant. The S 1 extension Let e: c t denote the integral lattice of T in t (i.e., the kernel of exp : t--+ T) and write 2~ tZ = { l1r II I II E tZ} . Here is the central result of the chapter: 12.2 Theorem (S 1 Extension Theorem) Assume in addition to the hypotheses out- lined above the following (replacing F with an appropriate T-invariant subset if necessary): l. Each symplectic leaf Eµ (µ E tj) is diffeomorphic to an open disk in JR. 2 . THE S 1 EXTENSION FIGURE 107 1. The reduced dynamics in F/T, as described in the hypotheses of Theorem 12.2. 2. There exists a submanifold Z C F /T of codimension two that is transversal to each leaf, and having the following property (see Fig. 1): Z intersects Eµ at a single point Yµ that is an equilibrium point of Xh[Eµ, of elliptic type. 3. Each 'punctured disk' E~ = Eµ, \ {yµ,} is foliated by nontrivial periodic orbits of Xh. Next, let fo E 2~ tZ and co E R be arbitrary, and Jet I : F /T --+ R be the unique function satisfying: 4. I(yµ,) = co+ (µ,fo) (µ E tj). 5. Each punctured disk E~ is foliated by (minimally) 21r-periodic orbits of X1, which, up to orientation preserving reparametrizations of time, coincide with the orbits of Xh. In other words, I[E~ is a classical 'action' for the Hamiltonian h[E~ (see Remark 12.3 below). Then: 6. X1op has a well-defined 21r-periodic fl.ow (t, x) i---+ <I>t(x): Rx F--+ F. THE S 1 EXTENSION 108 7. The action of 3 1 defined by (x E F) commutes with that of T. 8. The Hamiltonian H is 3 1 -invariant. 9. The 3 1 action is Hamiltonian, with momentum map Io p: F ➔ Lie(3 1 )* S:! JR. 10. The momentum map Io pis both 3 1 - and T-invariant. 11. The momentum map J : F ➔ t* is 3 1 -invariant. 12. The 3 1 action is free on F' =F\p - (Z). 1 In particular, one can extend the action of T to an action of T' (q, s) • x = s • (q • x) = q • (s· x) 51 51 =T x 3 by defining 1 ( (q, s) E T', x E F ) . This action has T'-equivariant momentum map J' : F ➔ t'* S:! t* x IR given by J'(x) (J(x), (Io p)(x)). Furthermore, His T'-invariant, T' acts freely on F', and the space (F', w, T', J') is geometrically integrable. 12.3 Remark Once the value of I on Z is fixed a la condition 4, its (unique) extension to a function satisfying condition 5 can be described explicitly: One takes ( y E (F/T)\Z, µ = j(y)) , where wµ denotes the symplectic structure on the leaf I:µ, and Ily C I:µ denotes the oriented two-manifold whose boundary is the Xh-orbit through y. The nontrivial part of the theorem is conclusion 6. Specifically, we show that the Xropinvariant submanifold p- 1 (Z) C Pis foliated by (possibly trivial) ·2rr-periodic orbits of Xrop, and that the other integrable curves of X lop are minimally 2rr-periodic. The latter fact is not obvious a priori. Indeed if we relax condition 4 and let ti---+ 7(t) be an integral curve of X1op , with 7(0) E F', then although p('Y(O)) = p('Y(2rr)) ('Y is mapped to an integral curve of X1 since pis Poisson) , it can happen that 7(2rr) = q-7(0) for some nontrivial q ET. This ASIDE: ON COMPUTING PHASES 109 q, which depends only on the curve,'= po 1, is known as the phase change (or holonomy) along the curve,'. It can be computed using the technology of Ehresmann connections on principal bundles (see below). Our task splits into three steps. In Step 1 below, we show by appealing to condition 4 that p- 1 (Z) is foliated by 21r-periodic orbits of Xlop· In Step 2, we show that all integral curves of X1 lying in the same leaf I:µ have identical associated phase changes. In Step 3, we use the results of Step 1, and a certain continuity argument, to show that these phase changes are in fact trivial. This will prove 6, and the remaining conclusions of the theorem will follow easily. Before beginning the proof, we need to make two preliminary observations. The first concerns phases, and the second relative equilibria (i.e., equilibria in F/T). Aside: On computing phases We assume the reader is familiar with the concept of an Ehresmann connection on a principal T-bundle. The reader in unfamiliar territory is directed to Kobayashi and Nomizu (1963). Let F and B be arbitrary smooth manifolds, and suppose that p: F ➔ B is a principal T-bundle, with T acting on F from the left. If X is a T-invariant vector field on F, then there exists a vector field XE on B such that X and XE are p-related. If t f-t '"YB(t) is an integral curve of XE, then every integral curve t f-t 1 (t) of X with initial condition 1 1 (0) E p- (,s(0)) covers 'YB in the sense that p( 1 (t)) = ,B(t). If 'YB is periodic with (minimal) period t 0 > 0, then 1 (t 0 ) = q- 1 (0) for some uniquely defined q ET. The element q is independent of the initial condition 1 (0) E p- 1 (,B (0)) and will be called the phase change of X along ,B. For a discussion of phases in Hamiltonian mechanics, and their origins, see Marsden et al. (1990). Let a beat-valued connection one-form on the bundle p: F ➔ B. A vector field X on F is horizontal if X J a = 0. If Y is a vector field on B , then we denote by yh the horizontal lift of Y, with respect to a. This is the unique horizontal vector field yh p-related to Y. This vector field is T-invariant. ASIDE: RELATIVE EQUILIBRIA AND 'LIMITING PHASES' 110 The following fact is well-known. For a proof see, e.g. Marsden et al. (1990, pp. 39- 49). 12.4 Theorem l. There exists at-valued two-form CT on B, called the curvature 1 of a, well-defined by where V and W are any locally defined vector fields on B such that V(y) = v and W(y) = w. 2. Let X be a horizontal T-invariant vector field on B, and XB the corresponding vector field on the base B (i .e., such that X~ = X ). Lett.--+ ,.YB (t) E B be a periodic integral curve of XB, whose image is the oriented boundary of some compact two-dimensional submanifold EC B. Then the phase change q of X along ,Bis given by q = exp l CT • Aside: Relative equilibria and 'limiting phases' Ifs : M-+ N is a submersion, we write s* : Ann ker Ts-+ T* N for the map that is JR- linear on each space Ann kerTxs (x EM), and that sends dx(f os) to ds(x)f, for any locally defined function f on N. Here Ann denotes annihilator. Let (F, w, T, J) be a Hamiltonian T-space as before, and let ~ : Ann ker Tj -+ t be the composition of j* : Ann ker Tj -+ T *tj with the map T*tj -+ t** ~ t that 'forgets the base point'. 12.5 Proposition If y E F/T is an equilibrium point of Xh (so that dyh E Ann ker Tj), then XH(x) = 77F(x) where 1 77 = ~( dyh) Conventionally curvature refers to the two-form da - a I\ a on F . In the Abelian case considered here this form is the pull-back by p of O'. ASIDE: RELATIVE EQUILIBRIA AND 'LIMITING PHASES' 111 Recall that T/F denotes the infinitesimal generator on F corresponding to T/ E t. 12.6 Remark If F is a locally defined function on tj such that dyh = dy(F o j) (such a function will always exist since dyh E AnnkerTj), then ~(dyh) = PROOF OF 12.5. 1~, whereµ =j(y). Let F and µ be as in the remark. Then, recalling that J = j op, we have dxH = dx(F o J) for all x E p- 1 (y). In that case, by the collective Hamiltonian theorem (see, e.g., Marsden and Ratiu (1994, Theorem □ 12.5.2)). 12. 7 Corollary ( On limiting phases) Let E i---+ % be a smooth 2 family of periodic orbits of Xh, limiting on some equilibrium pointy E F/T, in the sense that Jim %(t) = y , t -+0 for all t. Let tt denote the (minimal) period of 'Yt and assume that the limit exists. Then where qt ET denotes the phase change of XH along%· PROOF. Apply the preceding proposition and the continuity of integral curves with respect to initial data. 2 -y,(t) should be smooth with respect to€ (with t fixed) and t (with l fixed). □ PROOF OF THE S 1 EXTENSION THEOREM 112 Proof of the S 1 Extension Theorem We are now equipped to prove conclusion 6 of Theorem 12.2 in the three steps outlined above. Step 1. Each Yµ, E F/T (µ E tj) is an elliptic equilibrium point of X1, so that dyµI E Ann kerTj. It follows from hypothesis 4 of the theorem that dyJ = dyµ(F op) where F(µ) =co+(µ, fo). By Remark 12.6, we have ~(dyµI) = fo, so that X1op(x) = (fo)F(x) for all x E p- 1 (Yµ,), by Proposition 12.5 (with H and h in 12.5 replaced by Io p and I). Since µ E tj was arbitrary, and fo E }" {2\ we conclude that the integral curves of X1op lying in the X lop-invariant submanifold p- 1 (Z) C P are 2rr-periodic (although these orbits may have a smaller - even trivial - period). Step 2. Fix someµ E tj. By Noether's theorem, X1op is tangent to J- 1 (µ) = p- 1 (Eµ,)- Write J- 1 (µ)' = p- 1 (E~) and (F/T)' = (F/T)\Z. Since p : F --+ F/T is a principal T-bundle, and E µ, C F /T is a (regular) su bmanifold, the restriction is also a principal T-bundle (remember J- 1 (µ) is T-invariant since Tis Abelian). We shall compute phase changes of X1op along curves lying in E~ by constructing a connection on Pµ,: J- 1 (µ)'--+ E~. See Fig. 3. Observe that action-angle coordinates (in the purely classical sense) can be constructed in the two-dimensional system (E~, wµ,, hlE~)- Here wµ, denotes the symplectic structure on the symplectic leaf Eµ,, restricted to E~. By hypothesis 5 of the theorem, we may take the 'action' to be Iµ,= IIE~: E~--+ K A conjugate 'angle' 0µ, : E~ --+ S 1 can be obtained as the restriction to E~ of an S 1-valued map 0, defined on some open neighborhood j- 1 (U) n (F /T)' of E~ as follows (UC tj denoting some open neighborhood ofµ). There exists (for sufficiently small U) a codimension one submanifold SC j- 1 (U) n (F/T)' that is transversal to each orbit of X1 in the (Xi-invariant) set j- 1 (U) n (F/T)', and intersecting each such orbit at precisely one point (see Fig. 2). The flow of X1 is 21r-periodic, inducing an action of S 1 on F/T. We PROOF OF THE S 1 EXTENSION THEOREM 113 z t J* j-1 (U) - ---- ~------ u J ◄ ----------- FIGURE 2. Constructing 0 : j- 1 (U) n (F /T) '-+ 5 1 . define 0 :j- 1 (U) n (F/T)'-+ 5 1 by declaring 0(y) to be the uniquely defined element of 5 1 such that y = 0(y)·s 1 s, for some (uniquely determined) s E 5. For the sake of a lucid notation, we shall commit the usual sin by pretending that 0 is an JR-valued function. We leave it to the reader to verify at each step that this sin can be forgiven. Note that by construction Iµ, and 0µ, = 01:B~, constitute action-angle coordinates in every leaf (µ' E U). Therefore, they are conjugate in the sense that 12.8 (µ'EU) . The preimage under p of the open set j- 1 (U) n (F/T)', on which 0 is defined, is the open set J- 1 (U) n F'. This latter set is an open neighborhood of J- 1 (µ)' in F'. ATinvariant Ehresmann connection on the bundle Pµ : J- 1 (µ)' -+ :B~ is obtained by declaring the horizontal space Horx at x E J- 1 (µ)' to be the space PROOF OF THE S 1 EXTENSION THEOREM 114 T-orbit, p- 1 (y) XIop(x) =we,,(x) FIGURE 3. Computing phases using the connection on Pµ,: J- 1 (µ)--+ I:~. (see Fig. 3). This choice is valid since: (i) X0 0 p and X1op are T-invariant, (ii) X0ap(x) and X1op(x) are tangent to J- 1 (µ)' (by Noether), and (iii) The image of Horx under Tp is We let aµ, denote the connection one-form on Pµ, : J- 1 (µ)'--+ I:~ that is compatible with the above choice of horizontal spaces. Note that we cannot extend aµ, to a connection one-form on J- 1 (µ)--+ Eµ,, This is because the vector field X0JE~ cannot be extended to a vector field on Eµ, without a discontinuity at Yw (This is why we need the 'limiting phase' argument to be expounded in Step 3 below.) By construction, the vector field X1opJJ- 1 (µ)' is horizontal with respect to aµ,, We may therefore employ Theorem 12.4 to compute its phase changes. A simple observation is [X?ap, XJ0 p](x) = [X0op, X1op](x) = [X0, X1](p(x)) = X{0,I}(p(x)) (xEJ- 1 (U)nF'). PROOF OF THE S 1 EXTENSION THEOREM 115 Since 0 and I are conjugate (in the sense of 12.8), {0, I} = 0 on j- 1 (U) n (F /T)'. Therefore, by 12.4.1, the curvature of aµ is zero. It follows from 12.4.2 that the phase change of X1op has the same value along every orbit of Xr lying in the puncture disk E~. We denote this constant phase by qµ ET (as in Fig. 3). Step 3. Keep µ E tj fixed as in Step 2. The nontrivial periodic orbits of Xr lying in E~ can be parameterized by their energy relative to the energy at yµ: Let I f denote the orbit with %(0) E S whose image is the (l(yµ) + 1:)-level set of Iµ- Then the family E f-t If limits on the equilibrium point Yµ as in the hypotheses of Corollary 12. 7. The orbit % has period = 21r, for all E. Since ~(dyµI) = fo (see Step 1) and fo E 2~(!--, we conclude from Corollary 12.7, and the results of Step 2, that qµ = idy. This completes the proof of conclusion 6 of tf Theorem 12.2. If X is any T-invariant vector field on F, and t f-t <I>t the corresponding flow map, then <I>t(q • x) = q • <I>t(x) for all q ET, and for all t E IR for which <I>t(x) is defined (x E F). Since X1op is T-invariant, this proves 7. Since Xrop and Xr are p-related , it follows that p is S 1-equivariant, where S 1 acts on Fas in 7, and on F/T as described in Step 2. Therefore where we have used the fact that h is constant on the orbits of Xr, which follows from condition 5. This proves conclusion 8. Conclusion 9 is obvious. That Io p is T-invariant is trivial. That it is S 1-invariant follows from the same argument used to prove the S 1-invariance of H = h op above. So conclusion 10 holds true. Conclusion 11 follows from the S 1 -equivariance of p, the Xrinvariance of the leaves Eµ (µ E tj), and the formula J = j op. Condition 12 follows from the S 1-equivariance of p. The only nontrivial fact left to check is the T'-equivariance of J'. Since T' is Abelian, this amounts to checking that J' is T'-invariant. This follows from the fact that both J and Io p are both T- and S 1-invariant. This completes the proof of Theorem 12.2. THE NON-ABELIAN CASE 116 The non-Abelian case 12.9 Corollary Let (P, w, G, J) be a Hamiltonian G-space and H : P-+ R a G-invariant Hamiltonian. Assume the symplectic cross section (F, wp, T, JF) satisfies the hypotheses of Theorem 12.2, with the role of H in 12.2 being played by HIF. (In particular, assume that G acts freely and that P is dynamically integrable). Let F' and Io p be as in the theorem and define K =Io p and P' =G(F') P. Then there is a Hamiltonian action of S on P, 1 C with respect to which H is invariant, and which has as an S 1 -equivariant momentum map K : P -+ R the well-defined extension of K : F -+ R given by K(g · y) =K(y) (g E G, y E F) Moreover, this action of S 1 commutes with that of G (so that G x S 1 acts on P), is free on P', and P' is G x S 1 -integrable (i.e., geometrically integrable with respect to the extended symmetry group G x S 1 ). PROOF. Combine Lemma 8.17 (taking H = S 1 ) with Theorem 12.2. D CHAPTER 13 Concluding remarks Generalizations We conjecture that the estimates on the evolution of J(xt) described in the Introduction (and in detail in Chap. 7) hold under weaker hypotheses than the existence of action-group coordinates requires. Of course one still needs some kind of integrability assumption, such as 'reduced spaces are zero dimensional,' and one may need to assume uniform orbit and co-adjoint orbit types. Also, a convexity (or quasi-convexity) assumption will be necessary. To generalize our results to other orbit and co-adjoint orbit types will require more general Hamiltonian G-space normal forms. Moreover, one will need very concrete realizations of these spaces, as well as a good handle on the symplectic structure, equations of motion, etc. One will also need explicit (local) complexifications of the space and its associated structures. The classification of Hamiltonian G-spaces satisfying various conditions is still the subject of considerable research. See Guillemin et al. (1996) for a recent bibliography. While more general Hamiltonian G-space normal forms are known, realizations as concrete as action-group coordinates do not seem to have been worked out. In the best scenario, these realizations will be of the form considered in our abstract formulation of Nekhoroshev's theorem (Chap. 5), so that deriving Nekhoroshev estimates will again reduce to verifying our Assumptions A- C. This hope is not entirely fanciful, as a couple of examples will demonstrate. Recall that to apply the abstract Nekhoroshev-Lochak theorem one needs 'coordinates' of the form G x U, for some open subset U of a vector space t and some compact realanalytic manifold G. In particular, G does not need to be a Lie group. For example, G could be a quotient of groups Gi/G2. A complexification of Gin that case would be Gf /Gf 117 FOURIER SERIES 118 Consider for instance geodesic motions on S3 (relevant to the regularized Kepler problem). The phase space is T* S 3 and a symmetry group is SO (4), acting by the cotangent lift of the standard action on S3 C JR. 4 . If we remove the zero-section O from T* S3 , then all points in P = T* S3 \0 have orbit type (SO(2)) and co-adjoint orbit type (T), where TC SO(4) is a maximal torus (so that we still have J(P) C fl;eg but G does not act freely on P). The orbit space P/SO(4) is identifiable with (0,oo) and the orbit space projection P---+ (0,oo) is in fact a trivial bundle: P ~ SO( 4)/ SO(2) X (0, oo). So we may take G1 = SO(4), G2 = SO(2), and U = (0, oo). In the 1:1:1 resonance the phase space is JR. 6 ~ c3, and a symmetry group is SU(3). Removing the origin, one obtains a uniform orbit type (SU (2)) and co-adjoint orbit type (SU (2)), so that both our orbit type conditions fail. Nevertheless one has diffeomorphisms R 6 \ {0} ~ S 5 x (0, oo) ~ SU(3) / SU(2) x (0, oo), which is again of the form proposed above. In both the above examples one can prove that Assumption A of Chap. 5 (the existence of a 'period lattice') holds. It may be worth trying to verify the remaining Assumptions B and C. It is worth observing that in both examples the space U can be identified with some open wall of the Wey! chamber of the symmetry group . Fourier series Lochak's approach to Nekhoroshev estimates, as exploited here, conveniently avoids the use of Fourier series. For other kinds of perturbation analysis (e.g., proving the existence of 'whiskered tori' in the perturbed system) Fourier series may be unavoidable. In principle there should be no obstacle to implementing Fourier series in such cases. After all, the generalization of Fourier series on tori to series on compact connected groups is well-known (see, e.g., Brocker and tom Dieck (1985)). APPENDIX A Proof of the Nekhoroshev-Lochak Theorem (abstract form) This appendix is devoted to proving Theorem 5.9. To a large extent we follow Lochak's original arguments (Lochak (1992), Lochak (1993)). With the exception of some comments of our own on the role of convexity, we offer minimal motivation. For deeper insight, the reader is referred to Lochak's original papers, especially Lochak (1993). To keep our exposition as simple as possible, we have restrained from employing two 'technical tricks' that would have improved our estimates. The first, discovered by Neishtadt (1984) and implemented in Lochak and Neishtadt (1992), concerns the normal form calculation. The idea is that one should make the first coordinate change larger than the others. The second trick we could have used is to rescale the frequency (in the simultaneous approximation argument) so that its largest component is unity. This allows one to lower the dimension by one. See Lochak (1992) for details. Consider the setup described in Chap. 5 and restrict attention to a value of the parameter p* with the property that St(p*) = f_ n , for some T* > 0 and n E {Z (St = 'vh) . Then according to Assumption A, Gff x {p*} is foliated by T*-periodic solution trajectories of = h(p)). One of our first objectives is to derive a normal form for the perturbed Hamiltonian H = Ho + F in a neighborhood of G x {A} (Proposition A.16 XHo (Ho(g, p) below). We begin with some preliminaries: Averaging over periodic orbits Define A.l so that 'v w (p) = ,J,. n for all p E BP. By Assumption A, X w has a well-defined flow map (t, x) t----+ <I>tv(x) : IR x (Gff x BP) ➔ Gff x BP satisfying <I>;,tT• = <I>tv, and such that <l>W 119 LIE TRANSFORMS AND THE ITERATIVE LEMMA 120 maps a set of the form ca x {p} (0 ::;:; a ::;:; a, p E BP) onto itself. In particular <I>tv maps D'Y (p*, r) onto D'Y (p*, r) for any values of the parameters I and r, and is the identity map when t = T*. For u E A<C(D'Y(p*, r)), we define u, u,Iu E A<C(D'Y(p*, r)) by u==u-u, 1T• tu o <I>tvdt Iu = -l T* o A.2 Lemma l. u+{W,Iu}=u 2. IIIul1~·,r : ;:; tr* 11u11~·,r PROOF. The estimate A.2.2 is obvious. Regarding A.2.1, one computes {W,Iu} = - ddTiuo<I>wlT=O l d =--- T* dr 1 = -T* 1T. t(uo<I>tv)o<l>wdtl -o 7 O - 1T. t(-u d o <I>tv )dr 0 dt t ] T. + -1 1T• u o <l>wdt t = - -1 [tu o <l>w O T* T* 0 .• = -u+u □ Lie transforms and the Iterative Lemma In what follows we shall be interested in Hamiltonians of the form 1l = Ho+Z+N LIE TRANSFORMS AND THE ITERATIVE LEMMA 121 with Z = 0. The Hamiltonian 5.5 is of this form if we define 1-l = H, Z = P and N = P. In the method known as Lie transforms, one makes symplectic changes of coordinates using the time-one map associated with the Hamiltonian flow generated by some 'auxiliary' Hamiltonian. Indeed let x E Ac(D,,(p*, r)) be given, and suppose that Xx has a welldefined flow map (t, x) r-+ <I>~(x) : [-1 , 1] x D,,-s(p*, r) ➔ D,,(p*, r), for some o E (0, ,]. Then ¢ = <I>~ is symplectic. The following computations make use of Taylor's formula with integral remainder: 1-l o </> =(Ho+ Z + N) +(Hoo</> - Ho)+ ( (Z + N) o </> - (Z + N)) =(Ho+ Z) + N + (Wo </>- W) + ( (Ho - W) o </> - (Ho - W)) + ( (Z + N) o </> - (Z + N)) =(Ho+ Z) + (N + {W, x}) 1 + (1 - r){ {W, x}, x} 0 <I>~dT + ~1 (Ho - W) + ~1 (Z + N) ' 1 where A.3 Choosing x = IN and applying A.2.1 (with u = N), we obtain 1-l O </> = (Ho+ Z') + 1 1 (1 - r){ {W, x}, x} 0 <I>~dT +~1(Ho - W) + ~1(Z + N) , where Z' = Z + N (so that Z' = 0). By A.2.1, and our choice x = IN, {W, x} = N - N = - N , so that We may therefore write 1-l o </>=Ho + Z' + N', if we define LIE TRANSFORMS AND THE ITERATIVE LEMMA 122 where A.4 Summarizing: A.5 Lemma Let Z, NE A<C(D-y(P*, r)) and 0 < 5 ~ 1 be given, with Z = 0, and assume that the Hamiltonian vector field associated with x =IN has a well-defined flow map Then the Hamiltonian 1i =Ho+ Z + N is pulled back by the symplectic coordinate transformation </> = <I>~ to 1i o </> = H 0 + Z' + N' , where l. Z' = Z + N 2. N' (so that Z' = 0) , =!:::.1(Ho - W) + !:::.1(Z + N) - !:::.2N and where !:::.1, !:::.2 : A<C(D-y(P*, r))-+ A <C (D-y_o(p*, r)) are the operations defined by equations A.3, A.4. A.6 Remark The term !:::.1 (Ho - W) corresponds to what Lochak (1992) refers to as the 'frequency shift' contribution to the transformed Hamiltonian. Lemma A.5 is the basis for proving the following 'iterative lemma': A.7 Lemma (Iterative Lemma) Suppose Z, NE A <C (D-y(P*, r)), 0 ~ r ~ 1, 0 < 5 ~ 1 and E ~ 0 are given, with Z = 0, and assume that JJZ + NJJ~•,r ~ 3EE l. J[NJJ~•,r ~ 2EE LIE TRANSFORMS AND THE ITERATIVE LEMMA 123 Furthermore, assume that for some 11 ;_>,: 1 2. 3. and where 4. Then there exists a symplectic diffeomorphism <p from D'Y_ 0 (p*, r) into D'Y(p*, r) such that the Hamiltonian 1-l =Ho+ Z + N is pulled back to 1-l o ¢ = Ho+ Z' + N' , for some Z', N' E A c (D'Y_ 0 (p*, r)) with Z' = 0 and 5. 6. IIZ' + N'II~~~ ~ IIZ + Nll~•,r + IIN'II~~~ , ~ ! IINllp.,r II N'llp.,r 'Y-8 e 'Y Furthermore, for all (g',p') E D'Y_ 0 (p*, r) 7. ( (g' p) = ¢(g'' p') ) . The main point is that the fluctuating part N' of the transformed Hamiltonian 1-l o <p is less than the fluctuating part N of the untransformed Hamiltonian 1-l, by a factor of e. (We -------- call N the fluctuating part of 1-l =Ho+ Z + N because Ho+ Z = 0.) Note also that A.7.2 and A.7.3 are the first conditions involving the parameter r, which has been free until now (apart from the modest requirement 5.8). LIE TRANSFORMS AND THE ITERATIVE LEMMA PROOF. 124 In preparation for applying Lemma A.5, define x = IN E AIC(D-r(P*, r)). Assume that 5 satisfies A.8 Then Assumption B ensures that Xx has a well-defined flow map (t, x) I-)- <I>~(x) as in the hypotheses of Lemma A.5, and that this map satisfies (-1 :::; t :::;; 1) . A.9 Furthermore, if </> = <I>~, then Assumption B also tells us that for all (g', p') E D-y-8 (p*, r) the estimate A.7.7 holds. Applying A.2.2: A.10 1 llxll~·,r :::; 2_T* IINll~·,r T :::; T*EE = a-pT: E The requirement A.8 is therefore met if But this is guaranteed by the hypotheses A.7.2 and A.7.3. For u E AIC(D-r(P*, r)) one computes using the definition A.3, A.9 and Assumption C, Similarly one computes from A.4, A.9 and Assumption C, A.12 Also, by A.3, A.9 and Assumption C, THE HAMILTONIAN IN NORMAL FORM 125 Since and his (m, M)-convex on int BP :) int B;;15(p*) (see 5.1), we have so that we arrive at the estimate A.13 Lemma A.5 applies, and can now use A.5.2, A.13, A.11 and A.12 to compute using A.7.1 and A.IO using A.7.2 since l 1 ? 1 and r ( 1 Combining this with A.7.3 gives the estimate A.7.6. Using A.5.1, one also computes IIZ' + N'II~~; = IIZ +R +N'II~~; ( IIZ +RII~~; + IIN'II~~; = 11z +Nil~~;+ IIN'II~~; ~ IIZ + N[[~··r + IIN'II~~; , which proves A.7.5. □ The Hamiltonian in normal form We now return our attention to the Hamiltonian H = Ho+F (Ho(g, p) =h(p)) THE HAMILTONIAN IN NORMAL FORM 126 of Chap. 5, and in particular to its restriction to the domain D"l(p*, r) . We continue to leave, and r as free parameters. Lets~ 1 be given and denote by [s] E Zits integer part. We seek to compose [s] coordinate transformations ¢1, ... , ¢[s] using the Iterative Lemma A.7 (with f = IIFII /E) . At every step we will take o= ,/(2s) so that, assuming and the hypotheses A.7.2 and A.7.3 will always be satisfied . Write 1i 0 = H, zo = P and N° = P. A successful jth application of the It- erative Lemma will deliver a symplectic diffeomorphism <f) from D"l-h/( 2s) (p*, r) into = 1{J- 1 o<f) = Ho+zj+Nj, for some zj and ,-,,. =- 11zj + NJ ll "lp.,r and (3·J =- IINjllp.,r N j Wl'th z-j -- 0 • In that case ' wr1't1'ng '--'-J h/(2s) "f-h/(2s)' D"(-(j-l)"l/( 2 s)(P*, r) and anew Hamiltonian 1ij the inequalities A.7.5 and A.7.6 will imply A.14 (3j ( 1 -f3j-1 e . The composition cp = c/>1 o • • • o ¢[s] : D"l-[sh/( 2s) (p*, r) ➔ D,y(p*, r) will restrict to a map ¢ : D"l1 2 (p*, r) ➔ D"l(p*, r) (since [s]/(2s) ( 1/2). What needs to be checked is that the hypothesis A.7.1 of the Iterative Lemma holds at every step. That is, we must inductively prove that A.15 for all 1 ( j ( [s]. By construction ao ( IIFII = EE and f3o ( IIFII ~ 2 IIFII = 2EE. This anchors the induction. Suppose that A.15 (and hence A.14) holds for all j ~ n (n < [s]) . Then A.14 implies f3n ( e- nf3o ~ 2e-nEE ( 2EE and So A.15 also holds in the case j = n + 1, which completes the induction. MOVING COORDINATE SYSTEMS AND REDUCTION TO THE ZERO FREQUENCY CASE 127 We have G'[s] ::,;; G'o + (e- 1 + · · · + e-[sl),Bo ::,;; 3EE and ,B[s] ::,;; e-[s],Bo ::,;; e- 8 (2eEE), and summarize the preceding arguments as follows: A.16 Proposition (The Hamiltonian in normal form) Assume that O ::,;; r ::,;; 1 and s ~ l satisfy for some O ::,;; 1 ::,;; 1 and li > 0 (l2 is defined by A. 7.4). Then there exists a symplectic change of coordinates¢: D"Y; 2 (p*, r)-+ D"Y(p*, r) such that Ho¢= Ho+ Z + N , IIZ + Nll~ii ~ 3EE , IINll~ii : ,; e-s (2eEE) ( (g' p) = ¢(g'' p') ) . The first of the three hypotheses is just 5.8, repeated here for completeness. Moving coordinate systems and reduction to the zero frequency case To expose the role played by convexity 1 in establishing bounds on the perturbed dynamics, we prefer over Lochak's argument one based on moving coordinate transformations that we describe next. Proposition A.16 says that there exists a change of coordinates ¢ such that Ho¢ = Ho + Z + N, for some Z and N with Z = 0 and such that N is exponentially small with 1 Here we are thinking of the first estimate in 5.3, which we have yet to employ. MOVING COORDINATE SYSTEMS AND REDUCTION TO THE ZERO FREQUENCY CASE 128 respect to the parameters. Restrict attention to the real domain Do(P*, r) = G x Br 15 (p*), and suppose that D(p*) = 0 (D = 'vh, H 0 (g,p) = h(p)). Then the convexity of hand the Morse lemma imply that in a neighborhood of G x {p*} the level sets of Ho are diffeomorphic to G x Sk, where Sk denotes the unit sphere int, centered at p*. In particular these level sets are compact. Now IIZ + Nll(tr ~ 3EE (by A.16), so for E sufficiently small the level sets of Ho</> are also compact (in some neighborhood of G x {p*}) and energy conservation implies that integral curves of XHo</>, beginning sufficiently close to G x {p*}, remain y'E-close for all time. One attempts to convert the case D(p*) #- 0 to the zero frequency case just described by making an appropriate moving coordinate transformation. The following lemma shows how to compute solution curves in a Hamiltonian system, as they appear in a moving coordinate system, by computing solutions to an associated nonautonomous system. The proof of the lemma is elementary and left to the reader. A.17 Lemma (On moving coordinate transformations) Let 1i and W be differen- tiable functions on D 0 (p*, r) and assume Xw has a well-defined flow map (t, x) i----+ <I>tv(x): JR x Do(P*, r)-+ Do(P*, r). Suppose ti----+ Xt E Do(P*, r) is an integral curve of Xl--l and define Xi= (<I>tv )- 1 (xt) (i.e., Xi is the point Xt as seen by the 'moving coordinates' (<I>tv )tEIR)Then ti----+ Xi is an integral curve of the non autonomous system (X1-l,t )tEIR, where 1-ii is the time-dependent Hamiltonian defined by By ti----+ x~ being an integral curve of the nonautonomous system (XH;)tEIR, we mean that (t E JR) . Taking W(g, p) = w(p) = D(p*) • p and 1i =Ho</>= H 0 + Z + N (with D(p*) possibly non-zero), we have 1-i~ = (Ho - W) + Z +No <l>W MOVING COORDINATE SYSTEMS AND REDUCTION TO THE ZERO FREQUENCY CASE 129 (because Ho and Z are <I>tv-invariant). Or, defining Hb = Ho - W and Nt = No <I>tv, we may write We can write Hb(g,p) = h'(p) if we define h' = h - w. Then if S1' = "vh', we have S1'(p*) = 0. We are therefore in the zero frequency situation already described, with the important exception that we are dealing with a time-dependent perturbation Z + Nt. Time-dependence destroys energy conservation, and therefore stability, as we argued above. However, the smaller the magnitude of the time-dependence (i.e., of 8Ntf 8t), the closer to stability one expects to get. Indeed this is the case: A.18 Proposition Let Z, Nt: D 0 (p*, r)--+ JR. be differentiable functions (t E JR.) such that Z = 0 and such that t f-t Nt(x) is differentiable for all x in the interior of Do(P*,r). Let a ~ 0 be any number such that and assume that for some positive r 1 ~ r one has 1. Consider the time-dependent Hamiltonian (t E JR.) where Hb = Ho - W , and an integral curve t f-t (gt, Pt) of the non autonomous system where and MOVING COORDINATE SYSTEMS AND REDUCTION TO THE ZERO FREQUENCY CASE 130 Note that O < c~ < 1, since c7 = m/M ~ 1. Notice also of course that the 'stability time' to approaches oo as 118Ntf 8tllb*'r-+ 0. PROOF. Suppose then that IPo-P* I ~ c~pr1, with c~ as above. Then by the compactness of Do(P*, c~r1) = G x Bc~pr 1 (p*), the curve ti--+ (gt, Pt) E Do(P*, ri) is well-defined until it reaches the boundary of D 0 (p., ri). That is, there exists a time st 0 (s = ±1 , 0 < t 0 ~ oo) such that Write Xt = (gt, Pt). Then the analogue of energy conservation for nonautonomous Hamiltonians is the identity A.19 Recall that one proves this by applying the chain rule and computing Substituting Ht = Ho+ Z + Nt into A.19 and integrating leads to where h' = h-w. Since D 2 h' = D 2 h, we can apply Taylor's formula with integral remainder to conclude fo (1 - r)D h(po + rop)(op, op)dr = fo st0 0;T (xT)dr 1 A.20 2 - ( (Z + Nst 0)(Xst 0 ) - (Z + No)(xo)) - Dh'(po)op , where op = Psto - Po- We will arrive at a lower bound on to proving our proposition by bounding the LHS of A.20 from below and the terms on the RHS from above . Indeed, by the (m, M)-convexity of h, we have A.21 MOVING COORDINATE SYSTEMS AND REDUCTION TO THE ZERO FREQUENCY CASE 131 Furthermore, we have A.22 and A.23 A.24 Applying the estimates A.21-A.24 to A.20, we obtain so that the preceding estimate yields The reader can check (recalling that c~ = (v15 - 2)c7/2 and O < c7 ~ 1) that the coefficient of rif a is bounded from below by c7/8. Together with our hypothesis on ri, this yields □ NEKHOROSHEV ESTIMATES IN THE NEIGHBORHOOD OF THE PERIODIC ORBITS 132 Nekhoroshev estimates in the neighborhood of the periodic orbits We now combine our preceding arguments to deduce Nekhoroshev-type estimates on motions in a neighborhood of G x {p*}, which by assumption is foliated by periodic orbits of XH. Let r ~ 0 be given and define R = 4r/c~ ~ 4r (c~ E (0, 1) being the constant defined in A.18). Assume A.25 for some l 1 > 0. Then Proposition A.16 applies, with r replaced with R, delivering some sym plectic coordinate change </> : D'Y ; 2 (p*, R) ----+ D-y (p*, R). Let (110, 1go) E Do(P*, r) C Do(P*, R) be given. Then For t in some open interval of zero, there exists an integral curve t f-t (1gt, 1pt) E Do(P*, R) of XH with initial condition (1g 0 , 170) (remember r ~ R/4 < R). Our objective is to derive an estimate on the evolution of 11t· Define ,1..-l (1 ( 29t, 2Pt ) _ 'I-' 9t, 1Pt ) = • Then t f-t (2gt, 1Jt) is an integral curve of XHo</>· According to the last estimate in Proposition A.16, from which we deduce NEKHOROSHEV ESTIMATES IN THE NEIGHBORHOOD OF THE PERIODIC ORBITS 133 Choose 1 = c~. Then Define _ (;r.. t )-1 (2 2 ) ( 39t, 3Pt ) = '1.-'W 9t, Pt • By properties of <I>tv established at the beginning of this appendix, we known that 3p 0 = 2p0, so that the above estimate implies By Lemma A.17 and the arguments that followed, t 1-t (3gt, 3Pt) is a solution curve for the nonautonomous Hamiltonian where H~ = Ho - W, Nt =No <I>tv and where Z, NE D-y; 2 (p*, R) = Dc~; 2 (p*, R) are the functions delivered by Proposition A.16 (with r replaced with Rand 1 = c~). In particular, we know from A.16 that IIZ + Nll~{i~ ~ 3EE , IINll~{j~ ~ e- (2eEE) 5 Since No = N and IINtll~{j~ = IIN o <I>tvll:;;~ = IINll~{j~ (by forementioned properties of <I>tv), we deduce IIZ + Nollb.,R ~ 3EE , II a;t 11:··R = ll{Nt, W}llb.,R ~ C7~1 lrl(p*)I IINtll~,·i~ 4 7 ~ e-s ( c~:cTn EE) . 4 7 by Assumption C NEKHOROSHEV ESTIMATES IN THE NEIGHBORHOOD OF THE PERIODIC ORBITS 134 We now apply Proposition A.18 with a = 3E, r1 = 2r / c; = R/2 < R, and with r in A.18 replaced by R. To satisfy A.18.1 we requirer~ J12c; 2 aE/c 7 . The condition A.25 already guarantees this if the free parameter li ~ 1 is subject to the additional condition Since 1110 - P*I ~ 2pr = pc;r1, we can apply A.18 to the trajectory t t-----+ ( 3gt, 1Jt) to deduce where a ( suptEIR /E at o ) l ~llp.,R 2 ~ 3c; Tn s e 8ec4 Now 3pt = 1Jt, and by the last estimate in Proposition A.16 (with r replaced with R and 1 = c;), c' 2 1 I ,,.- 7R 1Pt - Pt ""' -4 p = pr In particular, so that Note that (1 + 2/c;)r ~ R/4 + R/2 < R. Summarizing: A.26 Theorem (c.f. Theorem 1, Lochak (1993)) Consider the Hamiltonian H =Ho+ F considered in Chap. 5 (H0 (g,p) = h(p)). Assume ESTIMATES FOR AN ARBITRARY INITIAL CONDITION 135 that Q(p*) = n/T*, for some p* E int B, n E {Zand T* > 0 (Q = 'vh). Furthermore assume that for some r 3 0, s 3 1 and l 1 3 max{l, 192a/c7 } we have where c~ = (v'5 - 2)c7/2 and l2 is given by A. 7.4. Then (real) integral curves t f--7 (gt, Pt) E G x B of XH satisfy the estimate cP□ ?•I ( r and ;~ ( :::>•) = IP,~ P,I ( (i + ;) r Taking s = cb/ 2 and r = (c~ /16)l 1 l 2 (TM /T*)i for some b > 0: 3 A.27 Corollary (c.f. Corollary 3, Lochak (1993)) Supposeli 3 max{l, 192a/c7} and b > 0 are given such that where and l 2 is given by A. 7.4. Then and Estimates for an arbitrary initial condition We now seek an estimate on solutions t f--7 (gt, Pt) with p0 = j5. Recall that j5 is a point in to, subject to the condition that the Hamiltonian is defined in some complex neighborhood ca X B 15 of G X {p}. ESTIMATES FOR AN ARBITRARY INITIAL CONDITION 136 Lochak's marvelous observation is that a stability time for any initial condition can be deduced from those of the periodic orbits studied above. To do so requires understanding how the periodic orbits are distributed, and the relevant tool is purely arithmetic. View {,8 1, ... , ,Bk} (the Z-basis for{?!.) as an JR-basis fort, and let,: JRk ➔ t denote the associated isomorphism. The 100 -norm on JRk induces a norm I · loo on t: ( Tj E JR) . We have 1(1 ~ c1v1kl(l 00 ((Et), courtesy of 5.7. Since ,(zk) = tz\ Dirichlet's simultaneous approximation theorem (see, e.g., Schmidt (1980)) tells us that z 1 (Vw E t)(VQ > 1)(:3q E [1, Q) n Z): lqw - t loo ~ Ql/k Taking w = O(p) and writing T* = TMq, one deduces: A.28 Theorem (On the proximity of periodic orbits) For all Q > 1 there exists T* > 0 and n* E ,J.. tZ such that Because of the convexity of h, the frequency map is locally invertible: A.29 Lemma Define c~ = min{c5 /2, l}. The map n: B ➔ t has a locally defined inverse n- 1 mapping BqcU( 2TM) (O(p)) diffeomorphically onto an open subset of B = B 15 (p). PROOF . Apply the inverse function theorem, as formulated in, e.g., Proposition 2.5.6 of Abraham et al. (1988) 2 . 2 □ Their statement contains a typographical error. min{(2K M)- 1 , R}, Q Correct values for the constants given are P =min{(2N L)- P/(2M)}, S =min{(2K M)- Q/(2L)}. 1 , 1 , = ESTIMATES FOR AN ARBITRARY INITIAL CONDITION 137 lfi- P*I = 1n- 1 (n(p)) - n- 1 (n*)I 11 = 11 1 Dn- (n* + t8D)8Ddtl ( 8n = D(fi) - n*) 1 = 1 [D h(n- (Q* + t8!l))]- 8!ldtl 2 1 1 1 :::; -18!11 = ,oTmlD(p) - n*1 . m With the aid of this estimate, the reader may readily verify that A.28 and A.29 combine to yield: A .30 Corollary Assume Then there exists T* > 0 and p* E B with !l(p*) E J.. t Z such that Let b > 0 and l1 ~ max{l, 192a/c7} be given. If we choose then the hypothesis of A.30 is satisfied if where [3 = min{c7c~, 2c1v'k}. In that case, for some T* > 0, with A.31 there exists p* E B such that !l(p*) E ,A tZ and A.32 IP - P*I 3 c~ l1l2 Eb "' 16 T*/TM ---~----- ,0 ESTIMATES FOR AN ARBITRARY INITIAL CONDITION 138 If we can satisfy the hypothesis on T*/TM in Corollary A.27, then for any solution t 1--r (gt, Pt) with Po= p we will be able to conclude from A.32 (using IPt - Pl ~ IPt - P*I + IP- P*I and T*/TM > 1) that A.33 11 ~ 3c;2 exp(E-b/2) 3--) c; 16l1l2 IPt - Pl $'. (2 + j5 "" c; ===} Tn, ' 8ec4 3 b E By A.31 the hypothesis on T*/TM in A.27 is met provided E A.34 A.35 and ~ ( ~ ) 1/b c; l1l2 El/2 -(l+k)b ~ l4l~+1/2 1 where We observe from A.31, A.32 and the definition of A*, that A ~ l - c; l1l2i 3 16 * 1/ So we ensure A* ? 1/2 if we assume u; C1'~1l,r • The inequality A.34 is guaranteed in that case provided E ~( ; ) 1/b . 2c~ li 12 What remains is to ensure A.35. Remember that 11 is still a free parameter, apart from the requirement 11 ? max{l, 192a/c7}. Choose 2 _ max { 1, 192a 11 = - , ( -1 )k+l/ } 14 C7 Then to satisfy the condition A.35 it suffices to ensure E1/ 2-(k+l)b ~ 1, which we achieve with the choice b= l 2(1 + k) ESTIMATES FOR AN ARBITRARY INITIAL CONDITION 139 To summarize, the exponential estimate A.33 holds, with /1 and b as above, provided This completes the proof of Theorem 5.9 and its Addendum 5.10. APPENDIX B Proof that W is a slice Suppose G acts smoothly on a smooth manifold M. Recall that S is a slice at x E M if Sis a Gx-invariant submanifold of M (G x CG the isotropy subgroup at x) containing x and such that G XGx S-t M 1 [g , x] i--+ g • s is a diffeomorphism onto some open neighborhood of G · x . Here G Xax Sis the quotient (G x S) / G x, where G x acts on G x S according to h • (g, s) =(g h- h · x), and [g, s] =(g, s) 1, mod Gx. We call S a global slice if the image of the map above is M. Sufficient conditions for a Gx-invariant submanifold S 3 x to be a slice at x are: (i) S intersects G-orbits in a 'complementary fashion,' i.e., T 5 M = T 5 S E9 Ts(G • s) (s ES), and (ii) s ES, g E G and g • s E S together imply g E Gx. Let us now verify that the (open) Weyl chamber W in fl* is a global slice for the coadjoint action of G on fl;eg· Since each regular co-adjoint orbit intersects W in a unique point (see, e.g ., Corollary 1.8.3, Part 1), it follows that G(W) = fl;eg (so that if Wis indeed a slice, then it is a global slice) and (w E W , g E G , and g • w E W) ⇒ g = id . By the above, it remains only to show that W intersects co-adjoint orbits in a complementary fashion, i.e. , (w E W) . B.l 140 B. PROOF THAT W IS A SLICE 141 We claim, identifying the various spaces with subspaces of fl*, that Tw(G • w) = !:.L = Ann t B.2 B.3 for all w E W. Then B.l follows immediately from the direct sum decomposition see , e.g., Sect. 1, Part l. Since W C !: is open, B.3 is obvious. PROOF OF B.2. Since regular co-adjoint orbits are of type (T) (by, e.g., 1.8.1, 1.8.2 and 1.2.2 of Part 1), we know that dim Tw(G • w) = dim!: (w E WC .tn fl;eg)- To prove B.2 it therefore suffices to show that T w(G · w) C !:.L = Ann t. Elements f3 of T w(G • w) are of the form f3 = ade w for some~ E fl. But then for all T E t we compute since flw t (by, e.g., Corollary 1.8, Part 1). This shows that f3 E Ann t = .t.L. Since /3 E Tw(G • w) was arbitrary, this proves Tw(G · w) C !:.L, as required. □ APPENDIX C Proof of the Extension Lemma This appendix is devoted to pay proof of Lemma 8.8. Recall the fact (following from the last statement in Theorem 8.8) that every point of P is of the form g • y for some g E G and y E F. We shall use this fact repeatedly in the sequel without further comment. Let us check that the extended action of H is well-defined. As in the proof of Theorem 8.14, we have g'·y' = g·y (g,g' E G; y, y' E F) if and only if g' = gq and y' = q- 1 -y for some q ET . In that case, for any h EH, we obtain g' • (h-Hy') = (gq) • (h •H (q- 1 • y)) = g • (h·Hy), since the actions of T and H on F commute by hypothesis. This shows that the extended action is well-defined. That the extension of K is well-defined follows similarly, appealing to hypothesis 2. Next, we argue that the extended action of H is symplectic. Fix some h E H and define ¢ : P -+ P by ¢( x) = h·H x. We need to show that ¢ is sym plectic. Since G acts symplectically on P, and H acts symplectically on F, it follows from the definition of the extended action that ¢ maps fibers of 1r : P -+ 0 symplectically onto themselves. It therefore remains only to show that Tc/> maps horizontal spaces of the symplectic connection on 7r : P-+ 0 symplectically onto horizontal spaces. From the definition of¢, the extended action of H, and infinitesimal generators, it follows that C.l Tc/>. ~p(g. y) = ~p(cp(g. y)) (~ E g, g E G, y E F) Since J is G-equivariant and JIF is H-invariant (hypothesis 3), we compute = g. J(h-Hy) = g. J(y) = J(g. y) That Tc/> is symplectic on horizontal spaces now follows from C.1 and Lemma 8.12. 142 C. PROOF OF THE EXTENSION LEMMA 143 Conclusion 4 follows by construction. So does conclusion 5. Conclusion 6 was proven in C.2 above. It remains to prove that K : P -+ Q* is a momentum map for the extended action of H. Write ryft (resp. ry{f) for the infinitesimal generator of the H action on P (resp. F) corresponding to 7J E Q. Let 7J E Q be arbitrary. Then one computes using the definition of the extended action, (g E G, y E F) , C.3 where <I> 9 (x) = g · x (x E P). Consider the one-form a= ryft J w - dK.,,. Then as 7J E Q is arbitrary, proving that K : P -+ Q* is a momentum map for the extended action amounts to proving that a vanishes. It suffices to check C.4 and C.5 aiflp(g • y) = 0 , where g E Gandy E Fare arbitrary, and flp(x) = Tx(G • x) (x E P). Suppose v E T 9 .yFg•µo· Then v = T<I> 9 • w for some w E TyF. In that case we compute (a, v) = w ( ryft (g · y), T<I> 9 • w) - (dK.,,, T<I> 9 • w) = w ( T<I> 9 • ry{f (y), T<I> 9 • w) - (dK.,,, w) = w(ry{f (y), w) - (dK.,,, w) using C.3 and the G-invariance of K since G acts symplectically = 0' since K : F -+ Q* is a momentum map for the action of H on F, by hypothesis. This proves C.4. Let ~ E fl be arbitrary. Then, using C.3, the G-invariance of K, and the symplecticity of <I> 9 , one computes (a,~p(g · y)) = (a, T<I> 9 • (g- 1 • ~)p(y)) C.6 = W ( ryff (y), (g- 1 • ~)p(y)) - (dI<.,,, (g-l. OP(Y)) C. PROOF OF THE EXTENSION LEMMA By Lemma 8.12.1, we can write g- 1 • ~ = 144 e + e for some e, e E g such that ~},(y) E Hory and ~i(y) E TyF, Then the right-hand side of C.6 becomes The first term vanishes, by the definition of Horx, The second term vanishes because K is G-invariant. The term in parentheses vanishes because K : F-+ !)* is the momentum map of the action of Hon F, and because ~i(y) E TF. Therefore, (o:,~p(g·y)) = 0. Since~ E g was arbitrary, equation C.5 must hold. APPENDIX D An application of the S 1 Extension Theorem: The axisymmetric Euler-Poinsot rigid body revisited The purpose of this appendix is to give an application of the results of Chap. 12. We show how the body symmetry of the axisymmetric Euler-Poinsot rigid body (viewed as a (S0(3), H)-integrable space) can be constructed ab initio by applying the 3 1 extension theorem (Theorem 12.2) . In principle, the same technique could be applied to obtain S0(3) x 3 1 symmetry in the asymmetric rigid body, provided one restricts to appropriate open subsets of the phase space. We do not attempt this here. In Chap. 11 we recalled that the Euler-Poinsot rigid body is described by the system (T S0(3), - de, H), where D.l and 0 is the one-form on T S0(3) defined by 11.2. Irrespective of the values of the moments of inertia 11 , f 2, h, the Hamiltonian H is invariant with respect to the action of G =SO (3) defined by =.\(AA, m) , D.2 A · .\(A, m) or equivalently by A · p(A, n) =p(AA, An) . (.\ and p are defined in Chap. 11.) This is just the action of S0(3) x 3 1 of Chap. 11 with the 3 1 part left out. A momentum map J : T S0(3) ---t IR 3 is given by J(.\(A, m)) = Am , D.3 or equivalently by J(p(A, n)) =n. 145 THE SYMPLECTIC CROSS SECTION 146 Let T SO(3) \0 denote T SO(3) with its zero section removed . Then G acts freely on T SO(3)\0 and J(T SO(3)\0) C fl;eg ~ JR. 3\{0}. The Hamiltonian G-space (T SO(3)\o, -de, SO(3), J) is (SO(3), H)- (i.e., dynamically) integrable. Before turning to the axisymmetric case, let us study the symplectic cross section of T SO(3)\0 in some detail. The symplectic cross section Choose T, W and O as in Example 8.6. Then W and O intersect at the point µ 0 = e3. The symplectic fibration 1r : T SO(3)\0 --+ 0 ~ 3 2 (see Theorem 8.8) is given by 1r(p(A,n)) = n/ llnll- The map 'I/;: SO(3) X (O,oo)--+ TSO(3)\0 defined by D.4 is a diffeomorphism onto the symplectic cross section F = 1r- 1 ( e3 ) of the space (T SO(3)\0, -de , SO(3), J) (see Definition 8.9). The map 'I/; is symplectic if we equip SO (3) x (0, oo) with the symplectic structure -de' , where e' is the one-form on SO(3) x (0, oo) defined by the formula 11.3. The action of T ~ 3 1 on SO(3) x (0, oo) that makes 'I/;: SO(3) x (0, oo) ~ F a T-equivariant map is given by D.5 The action of Ton F has momentum map JF: F--+ t* ~JR.given by JF(p(A, n)) = n · e3. So (JF o 1/;)(A,p) = p. To summarize, 'I/; : SO (3) x (0, oo) ~ F is an equivalence between the space (SO(3) x (O,oo),-de',31,JF o'I/;) and the symplectic cross section F, where e' and JF o 'I/; are as described above. THE SYMPLECTIC CROSS SECTION 147 Our next task is to obtain a concrete realization of the Poisson reduced space S0(3) x (0, oo)/3 1 . First, observe that the symplectic structure -d0' on S0(3) X (0, oo) is -w80 ( ) , 3 where w80 (3 ) is the symplectic structure that S0(3) x (0, oo) carries by virtue of being (up to the obvious identifications) the action-group model space G x to for G = S0(3) (see Lemma 9.4). This is no surprise since the symplectic structure on Gx to (G arbitrary) was defined precisely by realizing G X to as a symplectic cross section of T*G (Chap. 9) . The reason for the difference in sign observed here is the following: In the rigid body phase space T* S0(3) (which we have been identifying with T S0(3)) the symmetry group S0(3) of H acts by cotangent lifting the left action on S0(3) defined by g • h cotangent lifted the left action on G defined by g • h =hg- 1. =gh, while in Chap. 9 we In Example 10.8 we constructed action-group coordinates for geodesic motions on 3 2, i.e., we constructed an equivalence of Hamiltonian S0(3)-spaces ¢ : S0(3) x (0, oo) ~ T3 2 \0. This map, which was defined by is symplectic, when S0(3) x (0, oo) is equipped with the symplectic structure w80 (3 ). But as we are using S0(3) x (0, oo) as a realization of the symplectic cross section of T S0(3) \ 0, we are equipping S0(3) x (0, oo) with the symplectic structure -w80 (3 ). To make¢ symplectic we therefore redefine the symplectic structure on T 3 2\0 to be +de, where 0 is the one-form on T3 2 \0 defined in 8.13. We leave it to the reader to verify that the action of 3 1 on T 3 2 \0 with respect to which \J! is 3 1-equivariant is given by D.6 where µ =q xv/ llvll • The key observation at this point is that the orbits of this 3 1 action are the fibers of the map J : T3 2 \0 --t IR 3 \{0} defined by J(q , v) =q x v (not to be confused with the momentum map above of the same name). But we have already seen in Chap. 8 that J is the momentum map for a Hamiltonian action of S0(3) on T 3 2\O. Actually, since we have redefined the sign of the symplectic structure on T 3 2\O , the momentum map of this THE SYMPLECTIC CROSS SECTION 148 action is, in the current context, -J. In particular, since -J is equivariant, it is Poisson if we equip JR. 3 ~ .so(3)* with the positive Lie-Poisson bracket { ·, • }+ (see, e.g ., Marsden and Ratiu (1994, Chap. 10)). It follows that J is Poisson if we equip JR. 3 with the negative Lie-Poisson bracket { · , · }- = -{ ·, • }+- This structure is given by D.7 {J, h}_(y) = -y · V f(y) x Vh(y) . Since</>: SO(3) x (0, oo)-+ TS 2 \0 is symplectic, the composite p =Jo</>: SO(3) X (0, oo)-+ JR. 3\{0} is also a Poisson map. This map is also a surjection, whose fibers are the S 1-orbits in SO(3) x (0, oo) (by the S 1 -equivariance of</>). Thus (JR. 3 \{0}, { · , ·}-)is a realization of the Poisson manifold SO(3) x (0, oo)/S1, and p: SO(3) x (0, oo)-+ JR. 3 \{0} ~ SO(3) x (0, oo)/S 1 is a realization of the natural projection. Since HIF is T-invariant (T ~ S 1 ), the Hamiltonian 'lj;*H (which represents the restriction of H to the symplectic cross section) is S 1-invariant, dropping via p to a function on R 3 \{0}. Indeed, denoting the standard coordinate functions on JR. 3 by y1 , y2 , y3 , we have 'lj;* H = hop, where h : JR. 3\ {0} -+ R is given by The reader will readily verify this fact after observing that Hamiltonian vector fields on (JR. 3 \{0} , {·,·}-)take the form X1(Y) = -y x V f(y) , THE AXISYMMETRIC CASE 149 so that the equations of motion on IR 3\{0} corresponding to Hamiltonian hare • = ( -1 - -1 ) Y3Yl Y2 D.8 Ii h • Y3 = ( -1 - -1 ) Y1Y2 Ii h These are precisely the familiar Lie-Poisson reduced rigid body equations (see, e.g., op. cit.), albeit obtained via a rather convoluted route. Rather than Poisson reduce the full system directly, we have obtained these equations by doing Poisson reduction in (a realization of) the symplectic cross section. The axisymmetric case Before summarizing the above results, we need to restrict attention to an appropriate open subset of T SO(3)\0, in anticipation of applying Theorem 12.2. Assume the body is axisymmetric, Ii= h =I say. Then, writing z(t) =Y1(t)+iy2(t) E C, the general solution to the equations D.8 is D.9 D.10 D.11 where Points on the y3-axis or on the Y1-Y2 coordinate plane are equilibria. The remaining solutions are periodic with period (ay3 (0))- 1. The symplectic leaves of (IR 3\{0}, { •, • }-) are the spheres centered on the origin. The solution trajectories on a typical sphere are shown in Fig. 1. To apply Theorem 12.2 we need to restrict attention to the half-space lying above or below the Y1-Y2 coordinate plane. For concreteness, we choose the upper half-space IRt = {y E IR 3 I y3 > O}. Let us now summarize our previous results, with the various constructions restricted to the appropriate open subsets: THE AXISYMMETRIC CASE 150 equator FIG URE 1. The reduced dynamics of an axisymmetric rigid body on a typical symplectic leaf (sphere) . Points on the equator are equilibria. D.12 Proposition (Poisson reduction in the symplectic cross section) Define the open set PC TSO(3)\0 by and equip P with the symplectic structure w =-d0, where 0 is the one-form on T S0(3) defined by 11.2. Then P is invariant with respect to the action of S0(3) defined by D.2, which has momentum map J : P --+ JR. 3 defined by D.3. Define the open set S0(3)° C S0(3) by S0(3) where A33 =Ae3 • e3. Equip F' 0 = {A E S0(3) I A33 > O} , =S0(3) x (0, oo) with the symplectic structure w' = 0 -w80 (3 ), where w80 (3 ) is the symplectic structure carried by S0(3) x (0, oo) :) S0(3) 0 x (0, oo) by virtue of being (up to the obvious identifications) the action-group model space of S0(3) (see Lemma 9.4). Explicitly, w' = d0', where 0' is the one-form on S0(3) x (0, oo) given in Equation 11.3. Let T ~ S 1 act on F' according to D.5. This action has momentum map J' : F'--+ JR. defined by J' (A, p) = Then the map 7/J : F' --+ P defined by p. CONSTRUCTING THE S 1 EXTENSION IN THE SYMPLECTIC CROSS SECTION 151 D.4 is an equivalence from (F', w', T, J') onto the symplectic cross section (F, WF, T, JF) of (P, w, G, J). The map p : F'-+ JR.t defined by p(A, p) =pAe3 is a Poisson map onto (JR.t, { ·, • }-), wliere {·,·}-is defined by D. 7. Moreover, the fibers of p are the T-orbits in F', so that (l~t, { · , · }-) is a realization of the Poisson manifold F' /T and p is a realization of the natural projection F' -+ F' /T. Assuming 11 = fz we have H' ='lj;* H = hop, where h : JR.t -+ JR. is defined by =I, Constructing the 5 1 extension in the symplectic cross section Let (F', w', T, J') be the realization of the symplectic cross section F of P C T SO(3) given in the proposition above. The momentum map J' has image tj =(0, oo), and J' : F' -+ (0, oo) factors through p : F' -+ JR.t, delivering a map j' : JR.t -+ (0, oo) such that J' =j' op. Indeed j'(y) = IIYII- Each symplectic leaf Z:µ = (j')- 1 (µ) (µ E (0,oo)) is an open hemisphere of radius µ. If we take Z:::: {(0,0,t) It> O}, then the Hamiltonian T-space ( F', w', 1', J') satisfies hypotheses 1-3 of Theorem 12.2. The function I : F' /T ~ JR.t -+ JR. defined by I(y) = Y3 - n IIYII satisfies conditions 4 and 5 of 12.2 for any choice of integer n. We choose n =0 (because this choice will lead to the 5 1 extension corresponding to the body symmetry; a different choice leads to an equally valid, but different, 5 1 action 1 ). With this choice, we have Iop(A,p) = 1 For example, if we choose n = l, then l(y) = A/(2rr IIYII), where A denotes the area enclosed by the Xh-orbit parking through y (see Remark 12.3) . In that case, the function Io po 'lj;- 1 (extended by the S0(3) action to a function on P) corresponds to an 'action integral' for the rigid body appearing in Tantalo (1994). CONSTRUCTING THE S 1 EXTENSION IN THE SYMPLECTIC CROSS SECTION pA 33 . The corresponding function on the sym plectic cross section F is K 152 =Io po 1- One 1. computes (A E S0(3) 0 , p E (0, oo)) , where pis the map defined at the beginning of Chap. 8 (not to be confused with the map p giving us a realization of the quotient map F' -+ F' /T). By Corollary 12.9, we obtain a momentum map for the sought after 5 1 extension by extending K to a function on Pin the prescribed manner. Let A E S0(3) and m E IRt be arbitrary, and let us determine what the value of K(.X(A, m)) should be, according to the prescription of 12.9. Define A Then if g =exp((m x e3/ [[m[[)~) and p =[[ m[ [, so that Am = pe3. =AA- we compute using D.2 and 11.1, 1, .X(A, m) = g • .X(A, m) = g • p(A , Am)= g • p(A, pe 3) Therefore, we set K(.X(A, m)) =K(p(A,pe3)) = pA33 = [[m[[ ([;II) = m3 But this is precisely the momentum map for the familiar 5 1 action corresponding to the body symmetry (which in our case is symmetry about the e3-axis) . Since a Hamiltonian 5 1 action is determined by its momentum map, this proves our initial claim. APPENDIX E The Symplectic Leaf Correspondence Theorem Throughout this appendix 'smooth' means C 00 • Foliations with singularities The following discussion of generalized distributions, and their associated foliations, closely follows Libermann and Marie (1987 , Appendix 3). These notions were introduced and studied by Sussmann (1973), Stefan (1973) , and others 1 . A generalized distribution on a smooth manifold Mis a subset D of the tangent bundle TM with the property that D(x) =D n T xM is a subspace of T xM at every x EM. The adjective 'generalized' (which we will omit in the sequel) refers to the fact that we allow the dimension of D (x) to be x dependent . We write iJ for the set of locally defined smooth vector fields X on M satisfying X(x) E D(x) wherever X(x) is defined. A distribution D is deemed smooth if for all x E M there exist smooth vector fields X 1 , ... , X k E iJ defined in a neighborhood of x such that {X1 (x), ... , Xk(x)} is a basis for D(x). We are ultimately interested in the following special case: E.1 Example Let Q be a Poisson manifold and B the associated Poisson tensor, viewed as a vector bundle morphism B : T*Q--+ TQ. Next, define the characteristic distribution Don =B(T*Q ). If x1, ... , are local coordinate functions defined in a neighborhood of a point q E Q, then D(q) is generated by {Xx (q), ... , Xxj(q)}, where X1(x) =B(dxf). By Q by D Xn 1 removing appropriate vector fields if necessary, we may suppose that {Xx1 (q), ... , X xj (q)} is in fact a basis for D(q) . Since the Xxj are smooth and belong to iJ, the characteristic distribution D is smooth. 1 A vast bibliography on the subject of foliations has been supplied by Tondeur (1997) . 153 SYMPLECTIC LEAVES AND DUAL PAIRS 154 An integral manifold of a distribution D on a manifold M is an immersed connected submanifold I: C M such that T xI: = D(x) at all x E I:. A distribution D is integrable if through every point x E M there exists an integral manifold of D. E.2 Theorem and Definition Let D be a smooth integrable distribution on a smooth manifold M. Given x, y E M, write x containing both x and y. Then ~ y if one can find an integral manifold of D ~ is an equivalence relation on M, defining a partition of M called the (generalized) foliation associated with D. Equivalence classes of~ are immersed submanifolds of Mand are called leaves of the foliation associated with D. Each leaf is maximal in the sense that: (i) The leaf through a point contains all integral manifolds passing through that point; and (ii) No integral manifold properly contains any leaf. For a proof of E.2, see Libermann and Marle (1987, p. 385). If p: P ➔ Q is a smooth map, and Dis a distribution on Q, then we define a distribution p*D on P by declaring v E p*D(x) if Tp • v E D(p(x)). We call p*D the pull-back of D. Smooth integrable distributions, and their associated foliations, are well-behaved under appropriate pull-backs: E.3 Theorem Let p : P ➔ Q be a submersion and assume that the preimage of connected sets under p are always connected. Let D be a smooth integrable distribution on Q. Then the pull-back distribution p* D is smooth and integrable, and the leaves of the foliation on P associated with p* D are precisely those nonempty sets of the form p- 1 (I:), where Eis a leaf of the foliation associated with D. The proof of E.3 is postponed to the end of this appendix. Theorem E.3 will be crucial in the sequel. Symplectic leaves and dual pairs E.4 Theorem (Symplectic Stratification Theorem) The characteristic distribution on a Poisson manifold Q (see Example E.l above) is integrable. Furthermore, each leaf of the associated foliation inherits a symplectic structure THE SYMPLECTIC LEAF CORRESPONDENCE THEOREM 155 from the Poisson structure on Q in the following way: Given a realization i : S y Q of E (meaning that i is an injective immersion with i(S) = E), there exists a unique symplectic structure on S with respect to which i is a Poisson map. Recall that a smooth map of Poisson manifolds p: P-+ Q is Poisson if {J, h}op = {fop, ho p} for all locally defined functions J, h on Q. The leaves of the foliation in Theorem E.4 are called the symplectic leaves of Q. The above formulation of the Symplectic Stratification Theorem appears in Libermann and Marie (1987, p. 130). Its earliest incarnation is due to Kirillov (1976). E.5 Definition Let P be a symplectic manifold and Q 1 , Q 2 Poisson manifolds. A pair of Poisson maps Q 1 /:..!.__ P !!-3.+ Q2 is called a dual pair if ker T P1 and ker T P2 are symplectically orthogonal distributions. This pair is a full dual pair if P1 and P2 are surjective submersions. E.6 Example Suppose a Lie group G acts on Pin a Hamiltonian fashion, and let J : P-+ g* be a (not necessarily equivariant) momentum map. Then g* ? P -+ P/G is a dual pair. If G acts freely and properly, then the image gj of J, is an open subset of g*, and gj ? P-+ P/G is a full dual pair, with respect to an appropriate differentiable structure on P/G. The Symplectic Leaf Correspondence Theorem The main result we wish to recall in this appendix is a natural one-to-one correspondence between the symplectic leaves in each leg of a dual pair Q 1 /:..!.__ P !!-3.+ Q 2 . This result has been stated by Alan Weinstein, who sketched a proof under a slight variation of our hypotheses (Weinstein, 1983). We have been unable to fill in the details of Weinstein's original proof, however, and moreover we are unaware of a complete and correct proof in the literature. Although detailed proofs are probably known, we supply here for the record a detailed proof, which we hope is also correct. With Theorem E.3 in hand, the result follows quite naturally. We begin by understanding how p 1 and p2 pull back the characteristic distributions on Q 1 and Q 2 . THE SYMPLECTIC LEAF CORRESPONDENCE THEOREM 156 E.7 Lemma Let p: P ➔ Q be a Poisson submersion and let D denote the characteristic distribution on Q. If P is symplectic, and w denotes its symplectic structure, then p* D = ker T p + ( ker T p PROOF. Let Bp : T* P ➔ T P and BQ : T*Q t . ➔ TQ denote the Poisson tensors. Let p* : Ann ker T p ➔ T*Q denote the map sending dx (Jo p) to dp(x)f, for any locally defined function f on Q (Ann denotes annihilator). Then p: P ➔ Q being Poisson means BQ op*= Tpo Bp. Fix x E P. Then Here we have used the fact that p* maps Ann ker T p onto (in fact isomorphically onto) T;(x)Q. This follows immediately from its definition. The above calculation demonstrates that (Txp)- 1 (D(p(x))) = (kerTxp) + (kerTxp)w ( x E P), from which the claim of the lemma follows. □ If Q 1 ~ P !!3-+ Q2 is a full dual pair, and D 1, D2 denote the characteristic distributions on Q1,Q2, then p';_D1 = p2D2, by Lemma E.7. Let us suppose that P1 and P2 satisfy the connectedness hypothesis of Theorem E.2. Then by that theorem, each leaf of the generalized foliation associated with Pi D 1 = p2D2 is simultaneously the preimage of a symplectic leaf in Q1 and a symplectic leaf in Q2 . This establishes a one-to-one correspondence between the symplectic leaves in p 1 (P) C Q 1 and the symplectic leaves in p 2 (P) C Q2. Since we assume the dual pair is full, p 1 (P) = Q 1 and p 2 (P) = Q 2 . This furnishes a proof of the result we are after: E.8 Theorem (Symplectic Leaf Correspondence Theorem) Let P be a symplectic manifold and Q 1 ~ P !!3-+ Q 2 a full dual pair. Assume that each leg Pj : P ➔ Qj (i = 1, 2) satisfies the property that preimages of connected sets are connected. Let Fj denote the DETAILS 157 set of symplectic leaves in Qj . Then there exists a bijection :F1 ➔ :F2 given by having inverse Keeping Example E.6 in mind, an immediate corollary is Lemma 12.J stated in our discussion on converting dynamic integrability into geometric integrability. Details We now turn to the proof of Theorem E.3. We make use of the following observation: E.9 Lemma Let p: P ➔ Q be a surjective submersion of smooth manifolds, and let S C P be a compact pathwise connected set. Then S has a connected open neighborhood U with the property that the restriction p : U ➔ p(U) is a locally trivial fiber bundle. PROOF. Since S is compact and smooth finite dimensional manifolds are locally com- pact, we can cover S by a finite number of open subsets U1, ... , UN, each having compact closure in P. Since Sis pathwise connected, there exists an (open) connected component U of the union UjUj that contains S. The component U will have compact closure, so that the restriction p : U ➔ p(U) will be proper. This finishes the proof, since any proper surjective submersion is a locally trivial fiber bundle; see, e.g., Bates and Sniatycki (1992). □ A PROOF OF THEOREM E.3. We first verify that p* Dis smooth. Let x 0 E P be given and let U be a connected neighborhood of x 0 such that the restriction p : U ➔ p(U) is a locally trivial fiber bundle. The existence of such a U is guaranteed by Lemma E.9 above. In fact, restricting U if necessary, we may suppose that U is a globally trivial fiber bundle. It follows that there exist vector fields X 1 , ... X k defined on U (restricting U if necessary) that are tangent to the fibers of plU (and therefore belonging to p*D) and are such that {X1 (x), ... , Xk(x)} is a basis for the tangent space of the fiber through x, for all x in some open neighborhood of x 0 . Since D is smooth, there exist (restricting U if necessary) DETAILS 158 vector fields Yi, ... , Ym (k + m = dim P) on p(U) such that {Y1(p(xo)), ... , Ym(p(x 0))} is a basis for D(p(x 0)). Equip the locally trivial bundle p: U ➔ p(U) with the structure of an Ehresmann connection, and use this connection to lift the vector fields Y1 , ... , Ym to horizontal vector fields Y1, ... , Ym defined on U. Since Tp • Yj(x) = Yj(p(x)) (x EU), it follows that Yj E p*D. By construction {X1 (xo), ... , Xk(xo), Y1(xo), ... , Ym(x 0 )} is a basis of p* D(x 0). Since xo E P was arbitrary, this proves that p* Dis smooth. We next demonstrate that p* D is integrable. Let x 0 E P be given, and let E C Q be an integral manifold of D passing through p( x 0). We may suppose that E is a regular submanifold. (If not, replace Eby i(U) where U C Sis an appropriate open set and i : Sc.....+ Q denotes some realization of the immersed submanifold E.) By the preimage theorem, the set p- 1 (E) is a (regular) submanifold of P. This submanifold (which is connected since E is, by our hypotheses on p) is an integral manifold of p* D. Since it contains x 0 E P, which was arbitrary, this establishes the integrability of p* D. It remains to show that the leaves of the foliation on P associated with p* D are preimages of leaves of the foliation on Q associated with D. Let Z C P be a leaf, and let x 0 E Z be arbitrary. Denote by E the leaf of the foliation on Q containing y0 = p(x 0 ). We need to show that Z = p- 1 (:B). We begin by showing p- 1 (:B) CZ. Let x E p- 1 (E) be given and define y = p(x) E E. Let i : S c.....+ Q be a realization of the immersed submanifold E, and let s0 , s E S be the points such that i(s 0) = y0 and i(s) = y. Since S is connected (E is an integral manifold) there exists a continuous curve in S, defined on some closed interval, joining s0 and s. Since the image of this curve is compact, there exists a connected open neighborhood U of this image having compact closure in S (the relevant argument already appearing in our proof of Lemma E.9). In particular, E' = i(U) C E is an embedded integral manifold of D containing y0 and y. By the preimage theorem, p- 1 (E') is a regular submanifold of P. This submanifold (which is connected since E' is, by our hypotheses on p) is an integral manifold of p* D, and contains both x 0 and x. Therefore x 0 and x belong to the same leaf, namely Z. In particular, x E Z. Since x E p- 1 (E) was arbitrary, this establishes the inclusion p- 1 (E) C Z . DETAILS 159 Finally, let us prove the reverse inclusion Z C p- 1 (E). Let x E Z be arbitrary. Let i : S Y P be a realization of the immersed submanifold Z, and let 1 : [O, 1] -+ S be a continuous curve joining s 0 and s, where i(so) = xo and i(s) = x. By Lemma E.9, there exists a connected neighborhood U of ib([0, 1])) in P such that the restriction p : U -+ p(U) C Q is a locally trivial fiber bundle. Let V C S be an open connected neighborhood of the compact set 1 ([0, 1]) that is small enough to guarantee that the integral manifold Z' = s(V) of p* D is a regular submanifold of P. For example, take V to be a neighborhood with compact closure, so that the restriction i : V -+ P is proper. The set Z' contains the image of the curve / = i o 1 , which joins x 0 and x. There exist numbers t 0 , ... , tm with 0 = to < t1 < · · · < tm = 1 such that each curve segment ,'([tj-l, tj]) (1 ~ j ~ n) is contained in the domain Uj C P of a fiber bundle chart <pj : Uj -+ Wj x F (Wj C Q, p- 1 (Wj) = Uj, F = p- 1 (xo)). We claim that x = /(tm) is contained in p- 1 (:E). We prove this by induction on the integer indexing the curve segments. Since /(to)= x 0 belongs to p- 1 (:E), the induction is anchored. Let 1 ~ j ~ m be given and suppose that ,'(tj_i) E p- 1(:E). We need to show that ,'(tj) E p- 1 (:E). The distribution p*D is represented on the connected chart image Wj x F by the distribution D'(w, J) = D(w) EB T1F. One easily argues that the leaves of this distribution are of the form E' x F, where E' is a leaf of the foliation on Wj C Q associated with DIWj, It follows that the leaves of the foliation on Uj C P associated with p* DIUj are of the form p- 1 (E'), where E' is again some leaf of the foliation associated with DIWj , Some connected component of the set Z' n Uj contains ,'([tj-l, tj]). Let us call this component Zj. Then Zj is clearly an integral manifold of the foliation on Uj associated with p* D. Therefore Zj is contained in some leaf of this foliation (by the maximality of the leaves), i.e., in some set of the form p- 1 (E'), where E' is a leaf of the foliation on Wj associated with DIWj, In particular, pb'(tj-i)) and pb'(tj)) must lie on the same leaf E' of the foliation on Wj associated with DIWj, But this leaf is certainly an integral manifold of the foliation on Q associated with D. Therefore, Pb'(tj)) and Pb'(tj-1)) lie on the same symplectic leaf in Q. This leaf is E since p(/(tj_i)) EE, by the inductive hypothesis. So DETAILS 160 ,'(tj) E p- 1 (:E), which completes the induction . It follows th at x = ,'(tm) lies in p- 1 (:E). Since x E Z was arbitrary, this shows that Z C p- 1 (:E). □ Bibliography Abraham, R. and Marsden, J.E. (1978). Foundations of Mechanics, 2nd edn, Addison-Wesley Publishing Co., Reading, Massachusetts. Abraham , R., Marsden, J.E. and Ratiu, T. (1988). Manifolds , Tensor Analysis, and Applications, Vol. 75 of Applied Mathematical Sciences, 2nd edn, Springer-Verlag, New York. Arnold, V . I. (1963). Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russ. Math. Surv. 18 (5): 9-36. Arnold, V. I., Kozlov, V. V. and Neishtadt, A. I. (1988) . Dynamical Systems III, Encycl. Math. Sci., SpringerVerlag, New York. Bates, L. and Sniatycki, J. (1992). On action-angle variables, Arch. Rational Mech . 120: 337-343. Benettin, G. and Fasso, F . (1996). Fast rotations of the rigid body: a study by Hamiltonian perturbation theory. I., Nonlinearity 9: 137-186. Benettin, G. and Galgani, L. (1979) . Lyapunov characteristic exponents and stochasticity, in G. Laval and D. Gresillon (eds), International Workshop on Intrinsic Stochasticity in Plasmas , Editions de Physique, Courtaboeuf, Orsay, France. Benettin, G., Ferrari, G., Galgani, L. and Giorgilli, A. (1982). An extension of the Poincare-Fermi theorem on the nonexistence of invariant manifolds in nearly integrable Hamiltonian systems, fl Nuovo Cim ento 72B: 137-147. Bredon, G. E. (1972). Introduction to compact transformation groups, Academic Press, New York. Brocker, T. and tom Dieck, T. (1985). Representations of compact Lie groups, Vol. 98 of Graduate Texts in Mathematics, Springer-Verlag, New-York. Camacho, C. and Neto, A. (1985). Geometric theory of foliations , Birkhiiuser, Boston. Chirikov, B. (1979) . A universal instability of many-dimensional oscillator systems, Phys. Rep. 52(5): 265. Coffey, S. L., Deprit , A. and Miller, B. R. (1986) . The critical inclination in artificial satellite theory, Celestial Mechanics 39: 365-406. Cushman, R . (1977) . Notes on topology and mechanics, Unpublished. Dazord, P. and Delzant, T. (1987) . Le probleme general des variables actions-angles, J. Differential Geometry 26: 223-251. 161 BIBLIOGRAPHY 162 Duistermaat, J. J. (1980) . On global action-angle coordinates , Communications on Pure and Applied Mathematics 33: 687-706. Fasso, F. (1995). Hamiltonian perturbation theory on a manifold, Celestial Mechanics and Dynamical Astronomy 62: 43-69. Giorgilli, A. and Skokos, C. (1997) . On the stability of the Trojan asteroids, Astron. Astrophys. 317: 254-261. Gotay, M. J. (1982). On coisotropic embeddings ofpresymplectic manifolds, Proc. Amer. Math. Soc. 84: 111114. Grauert, H. (1952). On Levi's problem and the imbedding of real-analytic manifolds, Annals of Mathematics 68: 460-472. Guillemin, V. and Sternberg, S. (1984) . Symplectic Techniques in Physics, Cambridge University Press. Guillemin, V., Lerman, E. and Sternberg, S. (1996). Symplectic Fibrations and Multiplicity Diagrams, Cambridge University Press. Hilgert, J ., Neeb, K .-H. and Plank, W . (1994). Symplectic convexity theorems and coadjoint orbits, Compositio Mathematica 94: 129- 180. Kirillov, A. A. (1976). Elements of the Theory of Representations, Vol. 220 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin. Kirwan, F. (1984). Convexity properties of the moment mapping III, Invent. Math. 77: 547-552. Kobayashi, S. and Nomizu, K. (1963) . Foundations of Differential Geometry, Wiley, New York. Kolmogorov, A. N. (1954). Dokl. Akad. Nauk. SSSR 98 ( 4): 527-530. English translation: in Lecture Notes in Physics v. 93, G. Casati and J. Ford Editors, Springer, 1979. Libermann, P. and Marie, C.-M. (1987). Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company. Lochak, P. (1992). Canonical perturbation theory via simultaneous approximation, Russ. Math. Surv. 47(6): 57-133. Lochak, P. (1993). Hamiltonian perturbation theory: periodic orbits, resonances and intermittency, Nonlinearity 6: 885-904. Lochak, P. and Meunier, C. (1988). Multiphase Averaging for Classical Systems, Vol. 72 of Applied Mathematical Sciences, Springer. Lochak, P. and Neishtadt, A. I. (1992). Estimates of stability time for nearly integrable systems with a quasi-convex Hamiltonian, Chaos 2 : 495-499 . Marie, C.-M. (1982). Sous-varietes de rang constant et sous-varietes symplectiquement regulieres d'une variete symplectique, C. R. Acad. Sc. Paris 295: 119- 122. BIBLIOGRAPHY 163 Marle, C.-M. (1983a) . Normal forms generalizing action-angle coordinates for Hamiltonian actions of Lie groups, Letters in Mathematical Physics 7: 55-62. Marie, C.-M. (1983b) . Sous-varietes de rang constant d'une variete symplectique, Asterisque 107- 108: 6986. Marsden, J. E. (1981). Lectures on Geometric Methods in Mathematical Physics, Vol. 37 of GEMS, SIAM. Marsden, J .E. (1992). Lectures on Mechanics, London Mathematical Society lecture notes series, Cambridge University Press. Marsden, J.E. and Ratiu, T. S. (1994). Introduction to Mechanics and Symmetry, Vol. 17 of Texts in Applied Mathemati cs, Springer-Verlag. Marsden, J. E. and Weinstein, A. (1974) . Reduction of symplectic manifolds with symmetry, Reports on Mathematical Physics 5: 121-130. Marsden, J. E., Montgomery, R. and Ratiu, T . (1990) . Reduction, symmetry, and phases in mechanics, Memoirs. Amer. Math. Soc . 88(436) : 1- 110. Meyer, K. R. (1973). Symmetries and integrals in mechanics, in M. Peixoto (ed.), Dynamical Systems, Academic Press, pp. 259-273. Mishchenko, A. S. and Fomenko, A. T. (1978). Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. 12: 113-121. Moser, J. (1962). On invariant curves of area preserving mappings of an annulus, Nachr. Acad. Wiss. Gottingen Math. Phys . [(J 11a (1): 1-20. Moser, J. (1965). On the volume elements on a manifold, Trans . Amer. Math. Soc. 120: 286-294. Moser, J. (1970). Regularization of Kepler's problem and the averaging method on a manifold, Communi- cations on Pure and Applied Mathematics 23 : 609-636. Moser, J . (1973) . Stable and Random Motions in Dynamical Systems with special emphasis on Celestial Mechanics, Princeton University Press. Neishtadt, A. I. (1984). On the separation of motions in systems with rapidly rotating phase, J. Appl. Math . Mech. 48 : 133-139. Nekhoroshev, N. (1972). Action-angle variables and their generalizations, Trans. Moscow Math. Soc. 26 : 180198. Nekhoroshev, N. (1977) . An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russ. Math. Surv. 32 (6): 5- 66. Noid, D. W., Koszykowski, M. L. and Marens, R. A. (1977). A spectral analysis method of obtaining molecular spectra from classical trajectories , J. Chem. Phys. 67(2) : 404. BIBLIOGRAPHY 164 Poincare, H. (1993). New Methods of Celestial Mechanics. 1. Periodic and Asymptotic Solutions, Vol. 13 of History of Modern Physics and Astronomy, American Institute of Physics. Edited and Introduced by D. L. Goroff. Schmidt, W . M. (1980). Diophantine Approximation, Vol. 785 of Lecture Notes in Mathematics, Springer, Berlin. Smale, S. (1967). Differentiable dynamical systems, Bull. Amer. Math. Soc. 73: 747-817. Stefan, P. (1973). Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. 29(3) : 699-713. Sussmann, H. J. (1973). Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180: 171-188. Tantalo, P. (1994). Geometric Phases for the Free Rigid Body with Variable Inertia Tensor, PhD thesis, University of California, Santa Cruz. Tondeur, P. (1997). Geometry of Foliations, Birkha.user. Weinstein, A. (1971). Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6: 329346. Weinstein, A. (1981). Neighborhood classification of isotropic embeddings, J. Differential Geometry 16: 125128. Weinstein, A. (1983). The local structure of Poisson manifolds, J. Differential Geometry 18: 523-557. Whitney, H. and Bruhat, F. (1959). Quelques proprietes fondamentales des ensembles analytiques-reels, Comment. Math. Helv. 33: 132-160.