Liouville–Arnold theorem

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}

History

The theorem was proven in its original form by Liouville in 1853 for functions on ${\displaystyle \mathbb {R} ^{2n}}$ 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.

Statement

Preliminary definitions

Let ${\displaystyle (M^{2n},\omega )}$ be a ${\displaystyle 2n}$-dimensional symplectic manifold with symplectic structure ${\displaystyle \omega }$.

An integrable system on ${\displaystyle M^{2n}}$ is a set of ${\displaystyle n}$ functions on ${\displaystyle M^{2n}}$, labelled ${\displaystyle F=(F_{1},\cdots ,F_{n})}$, satisfying

• (Generic) linear independence: ${\displaystyle dF_{1}\wedge \cdots \wedge dF_{n}\neq 0}$ on a dense set
• Mutually Poisson commuting: the Poisson bracket ${\displaystyle (F_{i},F_{j})}$ vanishes for any pair of values ${\displaystyle i,j}$.

The Poisson bracket is the Lie bracket of vector fields of the Hamiltonian vector field corresponding to each ${\displaystyle F_{i}}$. In full, if ${\displaystyle X_{H}}$ is the Hamiltonian vector field corresponding to a smooth function ${\displaystyle H:M^{2n}\rightarrow \mathbb {R} }$, then for two smooth functions ${\displaystyle F,G}$, the Poisson bracket is ${\displaystyle (F,G)=[X_{F},X_{G}]}$.

A point ${\displaystyle p}$ is a regular point if ${\displaystyle df_{1}\wedge \cdots \wedge df_{n}(p)\neq 0}$.

The integrable system defines a function ${\displaystyle F:M^{2n}\rightarrow \mathbb {R} ^{n}}$. Denote by ${\displaystyle L_{\mathbf {c} }}$ the level set of the functions ${\displaystyle F_{i}}$,

$${\displaystyle L_{\mathbf {c} }=\{x:F_{i}(x)=c_{i}\},}$$

${\displaystyle L_{\mathbf {c} }=F^{-1}(\mathbf {c} )}$

Now if ${\displaystyle M^{2n}}$ is given the additional structure of a distinguished function ${\displaystyle H}$, the Hamiltonian system ${\displaystyle (M^{2n},\omega ,H)}$ is integrable if ${\displaystyle H}$ can be completed to an integrable system, that is, there exists an integrable system ${\displaystyle F=(F_{1}=H,F_{2},\cdots ,F_{n})}$.

Theorem

If ${\displaystyle (M^{2n},\omega ,F)}$ is an integrable Hamiltonian system, and ${\displaystyle p}$ is a regular point, the theorem characterizes the level set ${\displaystyle L_{c}}$ of the image of the regular point ${\displaystyle c=F(p)}$:

• ${\displaystyle L_{c}}$ is a smooth manifold which is invariant under the Hamiltonian flow induced by ${\displaystyle H=F_{1}}$ (and therefore under Hamiltonian flow induced by any element of the integrable system).
• If ${\displaystyle L_{c}}$ is furthermore compact and connected, it is diffeomorphic to the N-torus ${\displaystyle T^{n}}$.
• There exist (local) coordinates on ${\displaystyle L_{c}}$ ${\displaystyle (\theta _{1},\cdots ,\theta _{n},\omega _{1},\cdots ,\omega _{n})}$ such that the ${\displaystyle \omega _{i}}$ are constant on the level set while ${\displaystyle {\dot {\theta }}_{i}:=(H,\theta _{i})=\omega _{i}}$. These coordinates are called action-angle coordinates.

Examples of Liouville-integrable systems

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 ${\displaystyle \mathbb {R} ^{2n}}$, its coordinates are often written ${\displaystyle (q_{1},\cdots ,q_{n},p_{1},\cdots ,p_{n})}$ and the canonical symplectic form is ${\displaystyle \omega =\sum _{i}dq_{i}\wedge dp_{i}}$. Unless otherwise stated, these are assumed for this section.

• Harmonic oscillator: ${\displaystyle (\mathbb {R} ^{2n},\omega ,H)}$ with ${\displaystyle H(\mathbf {q} ,\mathbf {p} )=\sum _{i}\left({\frac {p_{i}^{2}}{2m}}+{\frac {1}{2}}m\omega _{i}^{2}q_{i}^{2}\right)}$. Defining ${\displaystyle H_{i}={\frac {p_{i}^{2}}{2m}}+{\frac {1}{2}}m\omega _{i}^{2}q_{i}^{2}}$, the integrable system is ${\displaystyle (H,H_{1},\cdots ,H_{n-1})}$.

• Central force system: ${\displaystyle (\mathbb {R} ^{6},\omega ,H)}$ with ${\displaystyle H(\mathbf {q} ,\mathbf {p} )={\frac {\mathbf {p} ^{2}}{2m}}-U(\mathbf {q} ^{2})}$ with ${\displaystyle U}$ some potential function. Defining the angular momentum ${\displaystyle \mathbf {L} =\mathbf {p} \times \mathbf {q} }$, the integrable system is ${\displaystyle (H,\mathbf {L} ^{2},L_{3})}$.

• Integrable tops: The Lagrange, Euler and Kovalevskaya tops are integrable in the Liouville sense.

See also

• Frobenius integrability: a more general notion of integrability.
• Integrable systems

References