Liouville–Arnold theorem - Imported Knowlpedia Content

Statement

Let $(M,\omega)$ be a $2n$ -dimensional symplectic manifold, and let $F=(F_1,\ldots,F_n):M\to\mathbb R^n$ have pairwise commuting components whose differentials are independent along $F^{-1}(c)$ . If $\Lambda$ is a compact connected component of this regular fiber, the Liouville–Arnold theorem states that $\Lambda$ is diffeomorphic to $\mathbb T^n$ and has a saturated neighborhood carrying action-angle coordinates. In those coordinates the fibers of $F$ are the tori $I=\mathrm{constant}$ , and every Hamiltonian commuting with all $F_j$ depends only on the actions. Consequently, its flow is linear on each such torus Arnol'd, Chapter 10.

Why a torus appears

The involution relations make the Hamiltonian vector fields $X_{F_1},\ldots,X_{F_n}$ commute. Regularity makes them linearly independent and tangent to $\Lambda$ , so their flows define a locally free $\mathbb R^n$ -action on the fiber. Compactness forces the stabilizer to be a full lattice; hence $\Lambda\cong\mathbb R^n/\mathbb Z^n$ . Period integrals then produce action variables, while the commuting flows provide angles.

Consequences

Each regular compact fiber is a Lagrangian submanifold. For $H=H(I)$ , Hamilton's equations reduce to

\dot I=0,\qquad \dot\theta=\nabla_IH,

so the dynamics is periodic when the frequency vector has rationally related components and quasiperiodic otherwise. The theorem therefore converts the local dynamics near a regular invariant torus into constant-velocity motion.

Hypotheses and global scope

Independence of the differentials and pairwise Poisson commutation are both essential. Compactness cannot simply be omitted: a regular invariant level may instead be a cylinder or another quotient of $\mathbb R^n$ . The resulting coordinates are local around one regular fiber. A family of such local charts need not glue globally; period-lattice monodromy is a standard obstruction Duistermaat, §§1–3.

References

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapter 10, the Liouville theorem and action-angle variables.
J. J. Duistermaat, “On global action-angle coordinates,” Communications on Pure and Applied Mathematics 33 (1980), 687–706. DOI record. Relevant: §§1–3, local theorem and global obstructions.