On the influence of numerical preservation of invariants when simulating Hamiltonian relative periodic orbits

We study the structure of the error when simulating relative periodic solutions of Hamiltonian systems with symmetries. We identify the mechanisms for which the preservation, in the numerical integration, of the Hamiltonian and the invariants associated to the symmetry group, implies a better time behavior of the error. A second consequence is a more correct simulation of the parameters that characterize the relative periodic orbit.