Separation of variables for bi-Hamiltonian systems

TL;DR

This paper develops a geometric framework using omega-N manifolds to identify separation variables for Hamilton-Jacobi equations in bi-Hamiltonian systems, providing intrinsic tests and explicit procedures.

Contribution

It introduces intrinsic tests for separability in bi-Hamiltonian systems using omega-N manifolds and offers explicit methods to find separated coordinates.

Findings

Intrinsic tests for separability in bi-Hamiltonian systems

Explicit procedures for finding separated coordinates

Application to Gel'fand-Zakharevich systems

Abstract

We address the problem of the separation of variables for the Hamilton-Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called omega-N manifolds, to give intrisic tests of separability (and Staeckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the omega-N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel'fand-Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.