Variational Discretizations for Hamiltonian Systems

TL;DR

This paper develops variational integrators for second-order Hamiltonian systems, ensuring structure-preserving numerical solutions, and demonstrates their effectiveness through applications to the Kepler problem and analysis of conserved quantities.

Contribution

It introduces a novel construction of variational integrators based on splitting techniques and provides conditions for Lagrangian existence in second-order systems.

Findings

Effective preservation of the LRL vector in simulations

Equivalence of integrators to explicit symplectic methods

Successful application to the Kepler problem

Abstract

In this paper, we study the Lagrangian functions for a class of second-order differential systems arising from physics. For such systems, we present necessary and sufficient conditions for the existence of Lagrangian functions. Based on the variational principle and the splitting technique, we construct variational integrators and prove their equivalence to the composition of explicit symplectic methods. We apply the newly derived variational integrators to the Kepler problem and demonstrate their effectiveness in numerical simulations. Moreover, using the modified Lagrangian, we analyze the dynamical behavior of the numerical solutions in preserving the Laplace--Runge--Lenz (LRL) vector.