In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also n independent, Poisson commuting first integrals of motion, and the level sets of all first integrals are compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time.
Thus the equations of motion for the system can be solved in quadratures if the level simultaneous set conditions can be separated.
The theorem is named after Joseph Liouville and Vladimir Arnold.
[1][2][3][4][5]: 270–272 The theorem was proven in its original form by Liouville in 1853 for functions on
with canonical symplectic structure.
It was generalized to the setting of symplectic manifolds by Arnold, who gave a proof in his textbook Mathematical Methods of Classical Mechanics published 1974.
-dimensional symplectic manifold with symplectic structure
An integrable system on
, satisfying 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
the level set of the functions
is given the additional structure of a distinguished function
, the Hamiltonian system
can be completed to an integrable system, that is, there exists an integrable system
is an integrable Hamiltonian system, and
is a regular point, the theorem characterizes the level set
of the image of the regular point
: A Hamiltonian system which is integrable is referred to as 'integrable in the Liouville sense' or 'Liouville-integrable'.
Famous examples are given in this section.
Some notation is standard in the literature.
When the symplectic manifold under consideration is
, its coordinates are often written
and the canonical symplectic form is
Unless otherwise stated, these are assumed for this section.