Poisson brackets after Jacobi and Plucker

TL;DR

This paper develops symplectic and bi-Hamiltonian structures for systems related to Jacobi elliptic functions, generalizes them to higher dimensions using projective lines, and explores conditions for Poisson bracket compatibility.

Contribution

It introduces a new class of rank 2 Poisson brackets parametrized by projective lines, linking their compatibility to geometric intersections and satisfying Jacobi identity via Plücker relations.

Findings

Poisson brackets satisfy Jacobi only when Plücker relations hold

Compatibility of brackets corresponds to intersecting lines in projective space

Examples demonstrate the geometric conditions for Poisson structures

Abstract

We construct a symplectic realization and a bi-hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Pl\" ucker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.