Tulczyjew’s triplet for lie groups III: Higher order dynamics and reductions for iterated bundles

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.

Contact Dynamics versus Legendrian and Lagrangian Submanifolds

“…For higher order Lagrangian dynamics, the triple is upgraded in [17,34]. For physical theories where the configuration space is a Lie group, the triple is examined in a series of works [35,36,37,47]. The triple is examined for principal fiber bundles in [39,38].…”

Tulczyjew's Triplet with an Ehresmann connection I: Trivialization and Reduction

“…The reduced dynamics on the reduced tangent bundle G\TQ is studied under the realm of the Lagrange-Poincaré equations. If, in particular, Q = G then the Lagrange-Poincaré equations reduces to the Euler-Poincaré equations on the Lie algebra g. Accordingly, this paper may be considered as a generalization of [18,19,17], wherein the trivialization and reduction of the Tulczyjew's triple were studied in the case Q = G. Referring the reader to [10] for further details on the reduced dynamics, especially the vertical and horizontal variations by means of an Ehresmann connection, we remak that the horizontal-vertical decomposition of the Tulczyjew's triple we shall present in this note provides a proper setting for the Legendre transformations of the horizontal-vertical Lagrange-Poincaré equations studied in [9,10,47]. In other words, these works carry a motivational importance for the present paper.…”