Symplectic Groupoids for Poisson Integrators

TL;DR

This paper introduces a novel method for constructing Poisson integrators using local symplectic Lie groupoids, leveraging Hamilton-Jacobi equations and the Magnus formula within Poisson geometry.

Contribution

It presents a new approach to Poisson integration via symplectic groupoids and interprets solutions as Lagrangian bisections, incorporating the Magnus formula for analysis.

Findings

Constructed Poisson integrators from symplectic Lie groupoids.

Interpreted Hamilton-Jacobi solutions as Lagrangian bisections.

Applied Magnus formula for backward error analysis.

Abstract

We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, give Poisson integrators. We also insist on the role of the Magnus formula, in the context of Poisson geometry, for the backward analysis of such integrators.