Symplecticity properties of Euler-Maclaurin methods

In this note we consider the use of Euler-Maclaurin methods for the solution of canonical Hamiltonian problems. As a subclass of multi-derivative Runge-Kutta methods, these integrators cannot be symplectic, however they turn out to be conjugate symplectic. The numerical solutions provided by a conjugate symplectic integrator essentially share the same qualitative long time behavior as those yielded by a symplectic integrator. This aspect, along with an efficient evaluation of the derivatives, suggests that Euler-Maclaurin methods could play an interesting role in the context of geometric integration.