The adaptive Verlet method

We discuss the integration of autonomous Hamiltonian systems via dynamical rescaling of the vector field (reparameterization of time). Appropriate rescalings (e.g., based on normalization of the vector field or on minimum particle separation in an N-body problem) do not alter the time-reversal symmetry of the how, and it is desirable to maintain this symmetry under discretization. For standard form mechanical systems without rescaling, this can be achieved by using the explicit leapfrog-Verlet method; we show that explicit time-reversible integration of the reparameterized equations is also possible if the parameterization depends on positions or velocities only. For general rescalings, a scaler nonlinear equation must be solved at each step, but only one force evaluation is needed. The new method also conserves the angular momentum for an N-body problem. The use of reversible schemes, together with a step control based on normalization of the vector held (arclength reparameterization), is demonstrated in several numerical experiments, including a double pendulum, the Kepler problem, and a three-body problem.