Numerical Analysis for a Class of Variational Integrators

In this paper, we study a geometric framework for second-order differential systems arising in classical and relativistic mechanics. For this class of systems, we derive necessary and sufficient conditions for their Lagrangian description. The main objectives of this work are to construct efficient structure-preserving variational integrators in a variational framework. To achieve this, we develop new variational integrators through Lagrangian splitting and prove their equivalence to composition methods. We display the superiority of the newly derived numerical methods for the Kepler problem and provide rigorous error estimates by analysing the Laplace–Runge–Lenz vector. The framework provides tools applicable to geometric numerical integration of both ordinary and partial differential equations.