On the relation between the velocity- and position-Verlet integrators

TL;DR

This paper compares velocity- and position-Verlet integrators through Hamiltonian analysis, showing they produce identical trajectories for harmonic oscillators but differ fundamentally in nonlinear systems, confirmed by numerical simulations.

Contribution

It provides a detailed Hamiltonian-based comparison of the two integrators, highlighting their equivalence in linear systems and fundamental differences in nonlinear dynamics.

Findings

Identical second-order differential equations for harmonic oscillators

Fundamental differences in nonlinear systems' Hamiltonians

Numerical simulations confirm analytical results

Abstract

The difference and similarity between the velocity- and position-Verlet integrators are discussed from the viewpoint of their Hamiltonian representations for both linear and nonlinear systems. For a harmonic oscillator, the exact Hamiltonians reveal that positional trajectories generated by the two integrators follow an identical second-order differential equation and thus can be matched by adjusting initial conditions. In contrast, the series expansion of the Hamiltonians for the nonlinear discrete dynamics clearly indicate that the two integrators differ fundamentally. These analytical results are confirmed by simple numerical simulations of harmonic and anharmonic oscillators.