Separation of Variables for Bi-Hamiltonian Systems

TLDR: In this article, the problem of the separation of variables for the Hamilton-Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry has been addressed, where the separation variables are naturally associated with the geometrical structures of the manifold itself.

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 ωN manifolds, to give intrisic tests of separability (and Stackel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the ωN 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.