  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.
