Completely integrable Hamiltonian system - Imported Knowlpedia Content

Definition

Let $(M,\omega,H)$ be a Hamiltonian system with $\dim M=2n$ . It is completely integrable in the Liouville sense if there are smooth functions $F_1=H,F_2,\ldots,F_n$ that are in involution,

\{F_i,F_j\}=0\quad\text{for all }i,j,

and are functionally independent on an open dense subset of $M$ , meaning $dF_1\wedge\cdots\wedge dF_n\neq0$ there. The map $F=(F_1,\ldots,F_n):M\to\mathbb R^n$ is the integral map. Some authors require independence only near a specified regular level, so the intended global or local convention must be stated.

Liouville–Arnold theorem

If $c$ is a regular value of $F$ and a connected component $L$ of $F^{-1}(c)$ is compact, then $L$ is an $n$ -torus. A neighborhood of $L$ admits action-angle coordinates $(I_1,\ldots,I_n,\theta_1,\ldots,\theta_n)$ in which

\omega=\sum_i dI_i\wedge d\theta_i

and $H$ depends only on the actions. Consequently, the Hamiltonian motion is linear on each nearby invariant torus Arnol'd, Chapter 10.

Regular and singular fibers

At a regular point of $F$ , the commuting Hamiltonian vector fields $X_{F_i}$ are linearly independent and span a Lagrangian distribution tangent to the common level set. Critical points of $F$ produce singular fibers, to which the regular Liouville–Arnold normal form does not directly apply. Global action-angle coordinates may also fail even when all fibers under consideration are regular; monodromy is one obstruction Duistermaat, §§1–3.

Examples and non-examples

The harmonic oscillator on $\mathbb R^{2n}$ , with the $n$ one-dimensional oscillator energies as integrals, is completely integrable away from their singular locus. A family of $n$ conserved quantities that satisfies a nontrivial functional relation is not enough: its differentials are dependent. Likewise, independent first integrals whose Poisson brackets do not vanish do not establish Liouville integrability.

Conventions and scope

“Completely integrable” can refer to other notions, including noncommutative integrability or integrability by quadratures. This knowl uses Liouville integrability. The requirement $F_1=H$ may be replaced by asking that $H$ be a function of a complete involutive family; locally at regular points these formulations agree in the usual setup.

References

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapter 10, complete integrability 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: global obstructions to action-angle coordinates.