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.
