Lagrangian Dynamics on Matched Pairs

TL;DR

This paper develops a framework for Lagrangian dynamics on matched pairs of Lie groups, deriving Euler-Lagrange and Euler-Poincaré equations, and applies it explicitly to the group SL(2,C).

Contribution

It introduces a novel approach to Lagrangian dynamics on matched pairs of Lie groups and relates it to semi-direct product theory.

Findings

Derived Euler-Lagrange equations for matched pairs.

Formulated Euler-Poincaré equations on matched pairs.

Applied the theory explicitly to SL(2,C).

Abstract

Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler-Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler-Poincar\'e equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the group $SL(2,\mathbb{C})$.