Numerical Analysis for a Class of Variational Integrators

In this paper, we study a geometric framework for a class of second order differential systems arising in classical and relativistic mechanics. For this class of systems, we derive the necessary and sufficient conditions for their Lagrangian description. Using the discrete variational principle, we present a class of variational integrators by splitting the Lagrangian. We then prove that the newly derived numerical methods are equivalent to composition methods. When applied to the Kepler problem, these methods show superior performance over classical methods. By establishing the theory of modified equations and modified Lagrangians, we provide error estimates for the new variational methods. The analysis of the Laplace--Runge--Lenz (LRL) vector confirms the numerical performance.