Applications of the Tensor Calculus to Problems Arising in Dynamics

Author: Trabant, Edward Arthur

Year: 1947

Degree: Dissertation (Ph.D.)

Advisor: Michal, Aristotle D.

Committee Member: Unknown, Unknown

Option: Mathematics; Physics

DOI: 10.7907/a75n-2j49

Abstract

The theory of the application of the tensor calculus to conservative dynamical systems is well known. However, the results of the actual application to specific systems are not. The thesis concerns itself with this aspect of the subject.

The gyroscope or spinning top is first considered. Necessary and sufficient conditions are developed in order that the Riemannian geometric space described by the system shall have constant Riemannian curvature, be an Einstein space, have a zero curvature invariant, etc. It is found that a simple relationship must exist between the moment of inertia coefficients. It is shown that the space described by the gyroscope is a special case of a general class of Riemannian spaces having the above properties.

A second type of conservative dynamical system is considered. It is shown that with a suitable choice of moment of inertias the above properties hold and that the geometric space obtained reduces to the gyroscope space.

An investigation is made into the form of the components of the fundamental metric tensor in order that general Riemannian spaces of three, four, and n-dimensions, suggested by the dynamical systems themselves, shall be Einstein spaces, possess constant Riemannian curvature and zero curvature invariant, and may be mapped conformally on a flat-space.

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