Tulczyjew's Triplet for Lie Groups III : Higher Order Dynamics and Reductions for Iterated Bundles

TL;DR

This paper develops a geometric framework for higher-order dynamics on Lie groups using Tulczyjew's triplet, incorporating subgroup reductions and exploring the structures of iterated bundles with Hamiltonian and Lagrangian formalisms.

Contribution

It extends Tulczyjew's triplet to higher-order iterated bundles on Lie groups, enabling systematic reductions and unified treatment of Hamiltonian and Lagrangian dynamics.

Findings

Unified geometric description of higher-order dynamics on Lie groups.

Implementation of subgroup reductions in the iterated bundle framework.

Application of Hamiltonian and Lagrangian formalisms to complex bundle structures.

Abstract

Given a Lie group $G$, we elaborate the dynamics on $T^*T^*G$ and $T^*TG$, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space $TT^*G$, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics.