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 energy level set is 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.
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
.
Theorem
If
is an integrable Hamiltonian system, and
is a regular point, the theorem characterizes the level set
of the image of the regular point
:
is a smooth manifold which is invariant under the Hamiltonian flow induced by
(and therefore under Hamiltonian flow induced by any element of the integrable system).
- If
is furthermore compact and connected, it is diffeomorphic to the N-torus
.
- There exist (local) coordinates on
such that the
are constant on the level set while
. These coordinates are called action-angle coordinates.
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.
- Harmonic oscillator:
with
. Defining
, the integrable system is
.
- Central force system:
with
with
some potential function. Defining the angular momentum
, the integrable system is
.
J. Liouville, « Note sur l'intégration des équations différentielles de la Dynamique, présentée au Bureau des Longitudes le 29 juin 1853 », JMPA, 1855, p. 137-138, pdf