Symplectic algorithms for simulations of rigid body systems using the exact solution of free motion

TL;DR

This paper introduces symplectic integration schemes of second and fourth order for rigid body simulations that leverage the exact solution of free motion, ensuring conservation of momentum and improved accuracy and efficiency.

Contribution

The paper presents novel symplectic algorithms for rigid body simulation that treat translational and rotational motions uniformly using exact free motion solutions.

Findings

Second order scheme is stable and highly precise in conserving constants of motion.

Schemes outperform existing methods in accuracy and efficiency at moderate densities.

Fourth order scheme is more efficient than second order when using sufficiently small time steps.

Abstract

Elegant integration schemes of second and fourth order for simulations of rigid body systems are presented which treat translational and rotational motion on the same footing. This is made possible by a recent implementation of the exact solution of free rigid body motion. The two schemes are time-reversible, symplectic, and exactly respect conservation principles for both the total linear and angular momentum vectors. Simulations of simple test systems show that the second order scheme is stable and conserves all constants of the motion to high precision. Furthermore, the schemes are demonstrated to be more accurate and efficient than existing methods, except for high densities, in which case the second order scheme performs at least as well, showing their general applicability. Finally, it is demonstrated that the fourth order scheme is more efficient than the second order scheme provided the time step is smaller than a system-dependent threshold value.