Symplectic splitting of Hamiltonian structures and reduced monodromy matrices

In Hamiltonian systems monodromy and reduced monodromy matrices of periodic orbits are symplectic. A symmetry is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. A natural class to study are periodic orbits which are invariant under such involutions. Depending on whether the orbit is invariant under a symplectic or antisymplectic symmetry, its monodromy matrix satisfies special properties. In the symplectic case, the monodromy matrix admits a symplectic decomposition, depending on the symplectic involution's fixed point set. In the anti-symplectic case, the monodromy matrix is of special symmetric form, which was constructed by Frauenfelder-Moreno. Such monodromy matrices are very beneficial in applied problems. The goal of this paper is to design a general construction behind the reduced monodromy's symplectic splitting, by using the notion of Hamiltonian manifolds, and to give a general framework for monodromy and reduced monodromy matrices of invariant periodic orbits in Hamiltonian systems with symmetries. In addition, we show an application related to contact forms, especially to the spatial circular restricted three-body problem for energy level sets of contact type.& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).