Preliminary definitions
Let be a -dimensional symplectic manifold with symplectic structure .
An integrable system on is a set of functions on , labelled , satisfying
- (Generic) linear independence: on a dense set
- Mutually Poisson commuting: the Poisson bracket vanishes for any pair of values .
The Poisson bracket is the Lie bracket of vector fields of the Hamiltonian vector field corresponding to each . In full, if is the Hamiltonian vector field corresponding to a smooth function , then for two smooth functions , the Poisson bracket is .
A point is a regular point if .
The integrable system defines a function . Denote by the level set of the functions ,
or alternatively, .
Now if is given the additional structure of a distinguished function , the Hamiltonian system is integrable if can be completed to an integrable system, that is, there exists an integrable system .