Collective symplectic integrators

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 $\mathfrak{so}(3{)}^{\ast }$. 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.

References in corpus (1)

• Collective Lie-Poisson integrators on ${\mathbf{R}}^3$

Cited by in corpus (3)

• Lie-Poisson methods for isospectral flows
• Geometry of discrete-time spin systems
• Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere