Reduction in mechanical systems with symmetry

TL;DR

This paper presents a global approach to reducing mechanical systems with symmetry, extending classical methods like Routh's, and applies it to rigid body motion, revealing new structural insights and equivalences.

Contribution

It introduces a global reduction method for mechanical systems with symmetry, generalizing classical local techniques and applying it to rigid body dynamics.

Findings

Global reduction reveals the structure of configuration manifolds.

The reduced system is equivalent to a particle moving on an ellipsoid.

Complete proof of Kolosov's theorem on rigid body reduction.

Abstract

The first part of the article is, in fact, the classical Routh method delivered in the language of contemporary theory of Lagrangian systems. But the Routh method deals only with concrete equations and, therefore, can be applied only in the case when the configuration spaces of the initial and the reduced systems are open submanifolds in Euclidean spaces. The global approach gives a possibility to find the structure of these manifolds in the general case and also to reveal some properties of the reduced system, first of all, the existence for this system of a global Lagrange function. We use the notion of a mechanical system introduced by S. Smale. The described method is applied to the global reduction in the problem of the motion of a rigid body having a fixed point in the potential force field with an axial symmetry. We present the complete proof of the theorem formulated by G.V. Kolosov on the equivalence of the reduced system in this case to the problem of the motion of a material point over an ellipsoid and also some corollaries of this theorem based on the results of L.A. Lyusternik and L.G. Shnirelman.