Collective symplectic integrators
Robert I McLachlan, Klas Modin, and Olivier Verdier

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 whose range is the target Lie--Poisson manifold, and their Hamiltonian is collective, that is, it is the target Hamiltonian pulled back by . The method yields, for example, a symplectic midpoint rule expressed in 4 variables for arbitrary Hamiltonians on . The method specializes in the case that a sufficiently large symmetry group acts on the fibres of , and generalizes to the case that the vector space carries a bifoliation. Examples involving many classical groups are presented.
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