On the coordinate system-dependence of the accuracy of symplectic numerical methods

TL;DR

This paper investigates how the choice of coordinate systems impacts the accuracy of symplectic numerical methods in simulating Hamiltonian systems, revealing non-invariance issues and potential for accuracy improvement through transformations.

Contribution

It provides a systematic analysis of coordinate dependence in symplectic methods, including derivations of non-invariance and conditions for preserving integrals, with numerical examples demonstrating these effects.

Findings

Coordinate transformations can significantly affect symplectic method accuracy.

Modified Hamiltonian is not invariant under coordinate changes.

Order improvements are possible via specific coordinate transformations.

Abstract

Symplectic numerical methods have become a widely-used choice for the accurate simulation of Hamiltonian systems in various fields, including celestial mechanics, molecular dynamics and robotics. Even though their characteristics are well-understood mathematically, relatively little attention has been paid in general to the practical aspect of how the choice of coordinates affects the accuracy of the numerical results, even though the consequences can be computationally significant. The present article aims to fill this gap by giving a systematic overview of how coordinate transformations can influence the results of simulations performed using symplectic methods. We give a derivation for the non-invariance of the modified Hamiltonian of symplectic methods under coordinate transformations, as well as a sufficient condition for the non-preservation of a first integral corresponding to a cyclic coordinate for the symplectic Euler method. We also consider the possibility of finding order-compensating coordinate transformations that improve the order of accuracy of a numerical method. Various numerical examples are presented throughout.