Lagrangian Reduction on Homogeneous Spaces with Advected Parameters

TL;DR

This paper investigates the Euler-Lagrange equations for parameter-dependent Lagrangians on homogeneous spaces, focusing on invariance properties and the resulting Euler-Poincaré equations with advected parameters.

Contribution

It introduces a framework for analyzing Euler-Lagrange equations on homogeneous spaces with advected parameters, highlighting invariance properties and their implications.

Findings

Derived Euler-Poincaré equations with advected parameters

Analyzed invariance properties of the Lagrangian

Connected pullback Lagrangian to Lie group structure

Abstract

We study the Euler-Lagrange equations for a parameter dependent $G$-invariant Lagrangian on a homogeneous $G$-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group $G$, emphasizing the special invariance properties of the associated Euler-Poincar\'e equations with advected parameters.