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Symplectic integrators in the realm of Hofer's geometry

Symplectic integrators constructed from Hamiltonian and Lie formalisms are obtained as symplectic maps whose flow follows the exact solution of a "sourrounded" Hamiltonian K = H + h^k H_1. Those modified Hamiltonians depends virtually on the time by h. When the numerical integration of a Hamiltonian system involves more than one symplectic scheme as in the parallel-in-time algorithms, there are not a simple way to control the dynamical behavior of the error Hamiltonian. The interplay of to different symplectic integrators can degenerate their behavior if both have different dynamical properties, reflected in the number of iterations to approximate the sequential solution. Considered as flows of time-dependent Hamiltonians we use the Hofer's geometry to search for the optimal coupling of symplectic schemes. As a result we obtain the constraints in the Parareal method to have a good behavior for Hamiltonian dynamics.