Reduction by symmetries of contact mechanical systems on Lie groups

2023/06/12 by Simoes, Alexandre Anahory, Colombo, Leonardo, de León, Manuel +3 · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2306.07028

Abstract

We study the dynamics of contact mechanical systems on Lie groups that are invariant under a Lie group action. Analogously to standard mechanical systems on Lie groups, existing symmetries allow for reducing the number of equations. Thus, we obtain Euler-Poincaré-Herglotz equations on the extended reduced phase space \mathfrakg× \R associated with the extended phase space TG× \R, where the configuration manifold G is a Lie group and \mathfrakg its Lie algebra. Furthermore, we obtain the Hamiltonian counterpart of these equations by studying the underlying Jacobi structure. Finally, we extend the reduction process to the case of symmetry-breaking systems which are invariant under a Lie subgroup of symmetries.

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• Structure-Preserving Optimal Control of Open Quantum Systems via a Discrete Contact PMP
• Variational Dissipative Mechanics on Lie Algebroids

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