A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems

By combining a standard symmetric, symplectic integrator with a new step size controller, we provide an integration scheme that is symmetric, reversible and conserves the values of the constants of motion. This new scheme is appropriate for long term numerical integrations of geodesic orbits in spacetime backgrounds, whose corresponding Hamiltonian system is non-integrable, and, in general, for any non-integrable Hamiltonian system whose kinetic part depends on the position variables. We show by numerical examples that the new integrator is faster and more accurate i) than the standard symplectic integration schemes with or without standard adaptive step size controllers and ii) than an adaptive step Runge-Kutta scheme.