Collective symplectic integrators

TL;DR

This paper develops symplectic integrators for Lie-Poisson systems using standard Runge-Kutta methods, enabling structure-preserving numerical solutions for complex Hamiltonian systems with symmetry.

Contribution

It introduces a novel class of symplectic integrators for Lie-Poisson systems based on collective Hamiltonians and standard Runge-Kutta methods, generalizing to systems with symmetries and bifoliations.

Findings

Constructed symplectic integrators for Lie-Poisson systems.

Provided explicit midpoint rule example for $ ext{so}(3)^*$.

Demonstrated applicability to classical groups.

Abstract

We construct symplectic integrators for Lie-Poisson systems. The integrators are standard symplectic (partitioned) Runge--Kutta methods. Their phase space is a symplectic vector space with a Hamiltonian action with momentum map $J$ whose range is the target Lie--Poisson manifold, and their Hamiltonian is collective, that is, it is the target Hamiltonian pulled back by $J$. The method yields, for example, a symplectic midpoint rule expressed in 4 variables for arbitrary Hamiltonians on $\mathfrak{so}(3)^*$. The method specializes in the case that a sufficiently large symmetry group acts on the fibres of $J$, and generalizes to the case that the vector space carries a bifoliation. Examples involving many classical groups are presented.