On bi-integrable natural Hamiltonian systems on the Riemannian manifolds

TL;DR

This paper introduces natural Poisson bivectors to unify and extend the construction of integrable Hamiltonian systems on Riemannian manifolds within bi-Hamiltonian geometry, broadening the scope of known integrable models.

Contribution

It generalizes the Benenti approach by defining natural Poisson bivectors, enabling a comprehensive framework for almost all known integrable systems on Riemannian manifolds.

Findings

Unified framework for integrable systems using natural Poisson bivectors

Extension of the Benenti approach to broader classes of systems

Connection of integrable systems with bi-Hamiltonian geometry

Abstract

We introduce the concept of natural Poisson bivectors, which generalizes the Benenti approach to construction of natural integrable systems on the Riemannian manifolds and allows us to consider almost the whole known zoo of integrable systems in framework of bi-hamiltonian geometry.