Reduction by Symmetries of Contact Mechanical Systems on Lie Groups

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-Poincar\'e-Herglotz equations on the extended reduced phase space \frakg \times \BbbR associated with the extended phase space TG \times \BbbR, where the configuration manifold G is a Lie group and \frakg 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. © 2024 Society for Industrial and Applied Mathematics.