Lie Groups and mechanics: an introduction

TL;DR

This paper introduces how Lie groups are applied in mechanics, covering rigid body motion, general Lie group theory, and specific applications to the diffeomorphism group with examples like the Burger and Camassa-Holm equations.

Contribution

It provides an accessible introduction to the use of Lie groups in mechanics, extending classical concepts to more general groups and specific fluid dynamics equations.

Findings

Lie groups effectively describe rigid body motion

Extension of Lie group theory to arbitrary groups in mechanics

Application to fluid dynamics equations like Burger and Camassa-Holm

Abstract

The aim of this paper is to present aspects of the use of Lie groups in mechanics. We start with the motion of the rigid body for which the main concepts are extracted. In a second part, we extend the theory for an arbitrary Lie group and in a third section we apply these methods for the diffeomorphism group of the circle with two particular examples: the Burger equation and the Camassa-Holm equation.