Numerical integration of variational equations

TL;DR

This paper compares numerical schemes for integrating variational equations in Hamiltonian systems, finding the tangent map method based on symplectic integration to be optimal in speed and accuracy.

Contribution

The paper introduces and evaluates various numerical schemes, highlighting the tangent map method as the most efficient for integrating variational equations in Hamiltonian systems.

Findings

Tangent map method outperforms other schemes in speed and accuracy.

Symplectic integrators effectively reproduce chaos indicators.

A systematic technique for constructing tangent maps is provided.

Abstract

We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the \textit{`tangent map (TM) method'}, a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton's equations of motion by the repeated action of a symplectic map $S$, while the corresponding tangent map $TS$, is used for the integration of the variational equations. A simple and systematic technique to construct $TS$ is also presented.