Symplectic Integrators in Corotating Coordinates

TL;DR

This paper develops three symplectic integrators for mass point dynamics in rotating coordinates, effectively handling non-canonical Hamiltonian systems with good energy conservation and long-term accuracy.

Contribution

It introduces three novel symplectic integrators tailored for non-canonical systems in corotating frames, including proofs of their properties and variational nature.

Findings

Integrators exhibit near-conservation of energy.

Numerical experiments show high precision and stability.

Methods are effective for complex rotating systems.

Abstract

The dynamic equation of mass point in rotating coordinates is governed by Coriolis and centrifugal force, besides a corotating potential relative to frame. Such a system is no longer a canonical Hamiltonian system so that the construction of symplectic integrator is problematic. In this paper, we present three integrators for this question. It is significant that those schemes have the good property of near-conservation of energy. We proved that the discrete symplectic map of $(p_n, x_n) \mapsto (p_{n+1}, x_{n+1})$ in corotating coordinates exists and the two integrators are variational symplectic. Two groups of numerical experiments demonstrates the precision and long-term convergence of these integrators in the examples of corotating top-hat density and circular restricted three-body system.