Second Order Lagrangian Dynamics On Double Cross Product Groups

TL;DR

This paper explores the geometric structure of second order Lagrangian dynamics on double cross product groups, deriving new equations and providing detailed formulations for these complex systems.

Contribution

It introduces a novel realization of the iterated tangent group as a double cross product, and derives second order Euler-Lagrange equations from first order equations in this context.

Findings

Derived second order Euler-Lagrange equations on the 2nd order tangent group.

Presented detailed formulation of second order Lagrangian dynamics.

Connected the structure of iterated tangent groups with double cross product groups.

Abstract

We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler-Lagrange equations on the 2nd order tangent group from the 1st order Euler-Lagrange equations on the iterated tangent group. We also present in detail the 2nd order Lagrangian dynamics on the 2nd order tangent group of a double cross product group.