Rigid Body with Rotors and Reduction by Stages

TL;DR

This paper compares various Lagrangian reduction methods for rigid bodies with rotors, highlighting their equivalences and conservation laws, to deepen understanding of symmetry reduction in mechanical systems.

Contribution

It introduces and compares different reduction procedures, including Euler-Poincaré reduction and reduction by stages, for systems with symmetry group $SO(3) imes S^1 imes S^1 imes S^1$.

Findings

Different reduction methods are shown to be equivalent under certain conditions.

The paper clarifies how conservation laws correspond across reduction procedures.

It provides a systematic comparison of reduction by stages and direct reduction techniques.

Abstract

Rigid body with rotors is a widespread mechanical system modeled after the direct product $SO(3)\times S^1\times S^1\times S^1$, which under mild assumptions is the symmetry group of the system. In this paper, the authors present and compare different Lagrangian reduction procedures: Euler-Poincar\'e reduction by the whole group and reduction by stages in different orders or using different connections. The exposition keeps track of the equivalence of equations as well as corresponding conservation laws.