Bi-Hamiltonian partially integrable systems

Abstract

Given a first order dynamical system possessing a commutative algebra of dynamical symmetries, we show that, under certain conditions, there exists a Poisson structure on an open neighbourhood of its regular (not necessarily compact) invariant manifold which makes this dynamical system into a partially integrable Hamiltonian system. This Poisson structure is by no means unique. Bi-Hamiltonian partially integrable systems are described in some detail. As an outcome, we state the conditions of quasi-periodic stability (the KAM theorem) for partially integrable Hamiltonian systems.

18 pages

References in corpus (5)

• The Poincare'-Lyapounov-Nekhoroshev theorem
• Geometric quantization of mechanical systems with time-dependent parameters
• Action-angle coordinates for time-dependent completely integrable Hamiltonian systems
• Geometric quantization of time-dependent completely integrable Hamiltonian systems
• Geometric quantization of completely integrable Hamiltonian systems in the action-angle variables

Cited by in corpus (10)

• Symmetries and Integrability
• Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds
• Noncommutative integrability on noncompact invariant manifolds
• Non-Noether symmetries in Hamiltonian Dynamical Systems
• Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system
• Systems of Hess-Appel'rot Type and Zhukovskii Property
• The Poincare'-Lyapounov-Nekhoroshev theorem for involutory systems of vector fields
• Quantization of noncommutative completely integrable Hamiltonian systems
• Geometrical aspects of integrable systems
• Superintegrable non-autonomous Hamiltonian systems