A symplectic integrator with arbitrary vector and scalar potentials

We study a new class of symplectic integrators for particles in arbitrary, time-dependent vector and scalar potentials. The methods were introduced in [Y.K. Wu, E. Forest, D.S. Robin, Phys. Rev. E 68 (2003) 046502] and are based on the ability to integrate Hamiltonians of the form (pi−ai(q))2(pi−ai(q))2 exactly for a finite time-step. We show that the integrators are symplectic in the non-relativistic case but not symplectic in the full six-dimensional phase space for relativistic Hamiltonians.