Lie-Poisson integrators
For researchers in geometric numerical integration, this paper provides a survey of methods for Lie-Poisson integrators, but it is largely a review without novel results.
This paper addresses the challenge of constructing geometric integrators for Hamiltonian systems on Lie-Poisson manifolds. It discusses the difficulties of standard techniques and presents modern approaches to preserve the Poisson structure and other fundamental properties.
In this paper, we discuss the geometric integration of hamiltonian systems on Poisson manifolds, in particular, in the case, when the Poisson structure is induced by a Lie algebra, that is, it is a Lie-Poisson structure. A Hamiltonian system on a Poisson manifold $(P, Π)$ is a smooth manifold $P$ equipped with a bivector field $Π$ satisfying $[Π, Π]=0$ (Jacobi identity), inducing the Poisson bracket on $C^{\infty}(P)$, $\{f, g\}\equiv Π(df, dg)$ where $f, g\in C^{\infty}(P)$. For any $f\in C^{\infty}(P)$ the Hamiltonian vector field is defined by $X_f(g)=\{g, f\}$. The Hamiltonian vector fields $X_f$ generate an integrable generalized distribution on $P$ and the leaves of this foliation are symplectic. The flow of any hamiltonian vector field preserves the Poisson structure, it fixes each leaf and the hamiltonian itself is a first integral. It is important to characterize numerical methods preserving some of these fundamental properties of the hamiltonian flow on Poisson manifolds (geometric integrators). We discuss the difficulties of deriving these Poisson methods using standard techniques and we present some modern approaches to the problem.