Liouville–Arnold theorem

When do conserved quantities turn a Hamiltonian problem into uniform motion on a torus? The Liouville–Arnold theorem requires enough independent, mutually commuting quantities and a compact, connected, regular common level. It gives action–angle coordinates near that torus. It does not turn every integrable trajectory into a closed orbit or supply one coordinate system through singular levels.

Required background. The Hamiltonian bracket bridge explains conservation, involution and canonical coordinates. The oscillator lesson constructs the simplest action and angle explicitly.

A regular compact common level

Let $(M,\omega)$ be a smooth symplectic manifold of dimension $2n$. Thus $\omega$ is a closed, nondegenerate two-form. We use $\omega=\sum_i dq_i\wedge dp_i$ in canonical coordinates and $\iota_{X_f}\omega=df$, so $X_f g=\{g,f\}$.

Take smooth functions $F=(F_1,\ldots,F_n)$, with $F_1=H$, that Poisson-commute on a neighborhood of a connected component $\Lambda$ of $F^{-1}(c)$. Assume $\Lambda$ is compact and

$$dF_1\wedge\cdots\wedge dF_n\ne0 \qquad\text{at every point of }\Lambda.$$

This is the regularity condition: the differentials are linearly independent. Then $\Lambda$ is a Lagrangian $n$-torus, and a sufficiently small neighborhood has coordinates

$$(\theta,I)\in\mathbb T^n\times B, \qquad \mathbb T^n=\mathbb R^n/(2\pi\mathbb Z)^n, \qquad \omega=\sum_{i=1}^n d\theta_i\wedge dI_i,$$

where $B\subset\mathbb R^n$ is open and the $F_i$ depend only on $I$. “Lagrangian” means that the restriction of $\omega$ to tangent vectors of the level is zero and its dimension is half that of $M$. The compact-component statement is the local regular-torus form of Cannas da Silva 2006 revision, § 18.4, Lemma 18.11 and Theorem 18.12, pp. 110–111, PDF, also established in Zung 2018, Theorem 2.1 and § 3.1, pp. 4–5 and 12–13, PDF.

The angles are circle-valued. The product description is global around this one torus after restricting the base $B$; individual real angle branches remain local. Compactness of this component is sufficient. The whole phase space need not be compact, and the map $F$ need not be globally proper.

Why the common level is a torus

The $X_{F_i}$ are independent, tangent to the level, and span its $n$-dimensional tangent space. Involution gives

$$\omega(X_{F_i},X_{F_j})=\{F_i,F_j\}=0, \qquad [X_{F_i},X_{F_j}]=-X_{\{F_i,F_j\}}=0.$$

Their flows are complete on the compact level. The resulting $\mathbb R^n$ action has open orbits, hence is transitive on the connected level. Its stabilizer is discrete by independence. Compactness makes that stabilizer a full-rank lattice, giving $\Lambda\simeq\mathbb R^n/\Gamma\simeq\mathbb T^n$.

To obtain canonical angles, normalize the nearby periods to a torus action preserving $\omega$. Contracting its generators with $\omega$ gives closed one-forms whose periods vanish: they vanish on the torus fibers, which carry the neighborhood’s first homology. Their potentials are independent actions. A remaining closed two-form on the base ball has a primitive and is removed by shifting the angles. This is a proof outline; the smooth construction and preservation argument are in Zung 2018, Theorem 2.2, Proposition 3.2 and equations (3.7)–(3.8), pp. 6–8 and 11–13, PDF.

Zung uses $\iota_X\omega_Z=-dH$ and angles $\phi$ of period one. Keeping the same physical $H$ and $X$, set $\omega_Z=-\omega$, $\theta=2\pi\phi$, and $I=z/(2\pi)$ for his action $z$. Both the sign and the period conversion are needed to obtain the displayed form above.

Linear motion is not necessarily a closed orbit

Because $H=H(I)$ in this neighborhood, Hamilton’s equations are

$$\dot I_i=0, \qquad \dot\theta_i=\Omega_i(I):=\frac{\partial H}{\partial I_i}.$$

Thus $\theta(t)=\theta(0)+\Omega(I)t$ modulo $2\pi$. Frequencies have units of inverse time when actions have units of action. A nonstationary orbit closes precisely when some $T\gt0$ satisfies $T\Omega\in2\pi\mathbb Z^n$. If no nonzero integer vector $k$ satisfies $k\cdot\Omega=0$, the orbit is dense in the full torus. Integer relations restrict it to a lower-dimensional subtorus; for more than two degrees of freedom, a resonance need not make the orbit periodic.

For two uncoupled oscillators with positive actions,

$$H=\omega_1 I_1+\omega_2 I_2, \qquad \omega_1,\omega_2\gt0.$$

The common level of $(I_1,I_2)$ is a two-torus. To use the theorem’s $F_1=H$ formulation, choose $F=(H,I_2)$; since $\omega_1\gt0$, this has the same levels and differential rank. The orbit is periodic if $\omega_1/\omega_2$ is rational, and dense if the ratio is irrational. For example, $(\omega_1,\omega_2)=(\omega_0,2\omega_0)$ closes after $2\pi/\omega_0$, while $(\omega_0,\sqrt2\,\omega_0)$ does not. The torus exists in both cases. No frequency nonresonance or nondegenerate Hessian of $H$ is required by the theorem.

The oscillator checks every hypothesis

For the harmonic oscillator,

$$H=\frac{p^2}{2m}+\frac{m\omega_0^2q^2}{2}, \qquad m,\omega_0\gt0,$$

an energy level $E\gt0$ is a compact connected ellipse. Its differential vanishes only at $(q,p)=(0,0)$, so that ellipse is regular. There is one integral for one degree of freedom, and $\{H,H\}=0$.

Following the physical orbit, define $I=(2\pi)^{-1}\oint p\,dq=E/\omega_0$. The parameterization

$$q=\sqrt{\frac{2I}{m\omega_0}}\sin\theta, \qquad p=\sqrt{2m\omega_0I}\cos\theta$$

gives $dq\wedge dp=d\theta\wedge dI$ and $\dot\theta=\omega_0$. In the $(q,p)$ plane, motion from the rightmost point goes downward: this physical orientation gives the positive action. At $E=0$, the ellipse collapses to a point and $dH=0$; the angle is undefined. The theorem cannot be continued through that singular fiber by pretending it remains a circle.

What fails without the hypotheses

Noncompact level. On $\mathbb R^2$ with $H=p^2/(2m)$, $m\gt0$, each component of $H^{-1}(E)$ for $E\gt0$ is a line $p=\pm\sqrt{2mE}$. It is regular, and the flow is complete, but $q(t)=q(0)+p(0)t/m$ is unbounded. Integrability alone has not produced a circle.

Insufficient independence. The pair $(H,H^2)$ does not supply two independent integrals: $d(H^2)=2H\,dH$. Counting formulas instead of differential rank can overstate what has been established. Independence must hold on the component being used, not merely at one sample point elsewhere.

Conservation without involution. Two functions may each commute with $H$ while failing to commute with each other. The two-oscillator calculation in the bracket bridge gives an explicit example. Such a pair does not meet the theorem’s hypotheses.

Noncompact Toda direction. In the full open Toda phase space, adding the same real constant to every $q_i$ preserves all differences and the Lax spectral invariants. Every common spectral level contains this unbounded translation orbit. The Toda proof establishes Liouville integrability, but its full levels do not satisfy this theorem’s compactness assumption. Fixing a center-of-mass frame would require a separate analysis of the remaining levels; it is not an automatic proof that they are compact.

Global coordinates. A local torus neighborhood does not prove that action–angle coordinates extend over an entire regular phase space, much less across critical values. Even the one-dimensional oscillator has no single continuous real-valued angle on its punctured phase plane, although its circle-valued angle is well defined there.

The pendulum lesson applies these hypotheses to librations, two distinct rotation components and the singular separatrix. Its transfer exercise keeps the local equations while changing the configuration space from a circle to the real line, making the role of global topology explicit.

Check the conclusion before using it

For three uncoupled oscillators with positive actions and frequencies $(\omega_0,2\omega_0,\sqrt2\,\omega_0)$, is a generic orbit periodic? Does the resonance destroy the three-torus of common action levels?

Solution

The relation $2\Omega_1-\Omega_2=0$ confines each orbit to a two-dimensional subtorus, with $2\theta_1-\theta_2$ fixed modulo $2\pi$. A return would require both $T\omega_0$ and $T\sqrt2\,\omega_0$ to be integer multiples of $2\pi$, which is impossible for $T\gt0$. The common action level is still a regular three-torus, equivalently the level of $(H,I_2,I_3)$; the closure of one orbit is smaller. This separates the level’s geometry from the arithmetic of its Hamiltonian frequencies.

References

• Cannas da Silva, Ana. Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764. Springer, 2001. DOI. Author revision January 2006, Open PDF. Page locators refer to that revision.
• Zung, Nguyen Tien. “A conceptual approach to the problem of action-angle variables.” arXiv:1706.08859v2 [math.DS], 4 February 2018. Version record. Open PDF.