The Jacobi-Maupertuis Principle in Variational Integrators

In this paper, we develop a hybrid variational integrator based on the Jacobi-Maupertuis Principle of Least Action. The Jacobi-Maupertuis principle states that for a mechanical system with total energy E and potential energy V{q), the curve traced out by the system on a constant energy surface minimizes the action given by / y^2{E — V{q))ds where ds is the line element on the constant energy surface with respect to the kinetic energy of the system. The key feature is that the principle is a parametrization independent geodesic problem. We show that this principle can be combined with traditional variational integrators and can be used to efficiently handle high velocity regions where small time steps would otherwise be required. This is done by switching between the Hamilton principle and the Jacobi-Maupertuis principle depending upon the kinetic energy of the system. We demonstrate our technique for the Kepler problem and discuss some ongoing and future work in studying the energy and momentum behavior of the resulting integrator.