Towards exact symplectic integrators from Liouvillian forms

Abstract: In this article we introduce a low order implicit symplectic integrator designed to follow the Hamiltonian flow as close as possible. This integrator is obtained by the method of Liouvillian forms and does not require particular hypotheses on the Hamiltonian. The numerical scheme introduced in this paper is a modification of the symplectic mid-point rule, it is symmetric and it is obtained by an isotopy of the deformation of the exact Hamiltonian flow to the straight line passing by two consecutive points of the discretized flow. This isotopy generates an alternative vector field on the flow lines transversal to the Hamiltonian vector field. We consider only the line arising from the mid-point to construct the symplectic integrator.