Time adaptive variational integrators: A space–time geodesic approach

The goal of this paper is to show that the space–time geodesic approach of classical mechanics can be used to generate a time adaptive variational integration scheme. The only assumption we make is that the Lagrangian for the system is in a separable form. The geometric structure which is preserved in the integration scheme is made explicit and the algorithm is illustrated with simulation for a compact case, a non-compact case, a chaotic system which arises as a perturbation of an integrable system and the figure eight solution for a three body problem.

► We show that standard Euler–Lagrange equations are equivalent to geodesic equations.
► The geodesic formulation enables time adaptive variational integrator for the dynamics.
► We demonstrate precisely in what sense the scheme is symplectic.
► We conclude with simulation results for integrable, chaotic and high dimensional three body example.