On bi-integrable natural Hamiltonian systems on the Riemannian manifolds

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.

24 pages, LaTeX with AMSfonts (some new references were added)

References in corpus (3)

• Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space
• Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices
• Separation of variables for the generalized Henon-Heiles system and system with quartic potential

Cited by in corpus (7)

• Simultaneous separation for the Neumann and Chaplygin systems
• New bi-Hamiltonian systems on the plane
• One invariant measure and different Poisson brackets for two nonholonomic systems
• On the Routh sphere problem
• Hamiltonization and separation of variables for Chaplygin ball on a rotating plane
• On one integrable system with a cubic first integral
• Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups