Hamiltonian and quasi-Hamiltonian systems, Nambu–Poisson structures and symmetries

Abstract

The theory of Hamiltonian and quasi-Hamiltonian systems with respect to Nambu–Poisson structures is studied. It is proved that if a dynamical system is endowed with certain properties related to the theory of symmetries then it can be considered as a quasi-Hamiltonian (or Hamiltonian) system with respect to an appropriate Nambu–Poisson structure. Several examples of this construction are presented. These examples are related to integrability and also to superintegrability.