Dedicated symplectic integrators for rotation motions

TL;DR

This paper develops specialized symplectic integrators tailored for rotation motions of rigid bodies, leveraging Lie algebra properties to improve efficiency and accuracy for asymmetric bodies in Hamiltonian simulations.

Contribution

It introduces a method to construct symplectic integrators with coefficients depending on moments of inertia, reducing stages needed for rigid body rotation simulations.

Findings

Dedicated integrators outperform general ones for highly asymmetric bodies.

Numerical tests show improved accuracy for specific cases.

Analytical estimates confirm limited advantages for symmetric bodies.

Abstract

We propose to use the properties of the Lie algebra of the angular momentum to build symplectic integrators dedicated to the Hamiltonian of the free rigid body. By introducing a dependence of the coefficients of integrators on the moments of inertia of the integrated body, we can construct symplectic dedicated integrators with fewer stages than in the general case for a splitting in three parts of the Hamiltonian. We perform numerical tests to compare the developed dedicated 4th-order integrators to the existing reference integrators for the water molecule. We also estimate analytically the accuracy of these new integrators for the set of the rigid bodies and conclude that they are more accurate than the existing ones only for very asymmetric bodies.