Action Integrals over Invariant Tori

In 1917 — in the middle of inventing general relativity's cosmology — Einstein wrote a short paper pointing out that the quantization rules of Bohr and Sommerfeld only make sense if the classical motion has a very particular geometry. The rule $\oint p\,\mathrm{d}q=nh$ integrates around a closed loop; but a generic bound trajectory never closes. What loop, then? Einstein's answer: the trajectory may not close, but for integrable systems it winds forever around an invariant torus in phase space, and a torus owns exactly $n$ independent, irreducible closed loops. The integrals belong to the loops of the torus, not to the orbit. That insight is the subject of this article. When a system with $n$ degrees of freedom has $n$ independent conserved quantities in involution, the Liouville-Arnold theorem confines each bounded motion to an $n$ -dimensional torus $T^n$ . On that torus the canonical one-form $\sum_ip_i\,\mathrm{d}q_i$ turns out to be closed, so its loop integrals do not care about the detailed shape of the loop — only its topological class. Integrating around the $n$ fundamental cycles produces $n$ numbers, the action integrals $J_i=\frac{1}{2\pi}\oint_{\gamma_i}\mathbf{p}\cdot\mathrm{d}\mathbf{q}$ : topologically protected constants that label which torus you are on. The actions are the natural coordinates on the space of tori — the momenta nature herself prefers. Everything downstream of this chapter is built on them: the angle variables and the transformation that trivializes integrable dynamics (Action-Angle Variables), the frequencies read off by differentiation (Frequencies from Action-Angle Variables), and the perturbation theory that asks which tori survive when integrability is broken.

Key takeaways

Bounded motion of a Liouville-integrable system with $n$ degrees of freedom lies on invariant $n$ -tori in phase space; the conserved quantities fix the torus, the trajectory winds around it forever.
A torus $T^n$ has exactly $n$ topologically distinct irreducible loops ( $H_1(T^n)\cong\mathbb{Z}^n$ ); the action integrals $J_i=\frac{1}{2\pi}\oint_{\gamma_i}\mathbf{p}\cdot\mathrm{d}\mathbf{q}$ are the canonical one-form integrated around each of them.
The invariant torus is a Lagrangian submanifold — the symplectic form vanishes on it — so the one-form is closed there and each $J_i$ depends only on the homology class of its loop, not on its shape.
The cycles are not trajectories: a generic orbit never closes, but the loop integrals are geometry, computed on the torus at frozen values of the conserved quantities.
For one degree of freedom, $2\pi J$ is simply the area enclosed in the $(q,p)$ plane; libration and rotation loops belong to different homology classes and carry genuinely different actions.
Key idea: on an invariant torus the momenta are functions of the coordinates and the canonical one-form $\sum_ip_i\,\mathrm{d}q_i$ is closed; integrating it around the $n$ irreducible cycles gives $n$ topologically protected constants $J_i=\frac{1}{2\pi}\oint_{\gamma_i}\mathbf{p}\cdot\mathrm{d}\mathbf{q}$ that label the torus and serve as the natural action momenta of the system.

The core insight

On an invariant torus $T^n$ the momenta are functions of the coordinates and the canonical one-form $\sum_i p_i\,\mathrm{d}q_i$ is closed; integrating it around the $n$ irreducible cycles gives $n$ topologically protected constants $J_i=\frac{1}{2\pi}\oint_{\gamma_i}\mathbf{p}\cdot\mathrm{d}\mathbf{q}$ that label the torus and serve as the action momenta of the system.

Where this lives today

The clearest modern habitat of invariant tori is a fusion device. Magnetic field lines in a tokamak or stellarator are the trajectories of a Hamiltonian system (one and a half degrees of freedom), and good confinement is the existence of invariant tori — the nested magnetic flux surfaces that keep hot plasma away from the wall. The actions are literally fluxes: the toroidal flux enclosed by a surface is its action variable, and "flux coordinates" used in every equilibrium code are action-angle coordinates under another name. Stellarator optimization — including the quasi-symmetric designs pursued through the 2020s — is in large part the engineering of a field whose line-flow has the largest possible measure of surviving tori, with island chains and chaotic layers (where actions cease to exist) squeezed to insignificance. The same mathematics runs the quantum-classical frontier under the name EBK (Einstein-Brillouin-Keller) quantization: semiclassical energy levels are assigned by demanding each action equal $(n_i+\mu_i/4)\hbar$ — Einstein's 1917 torus integrals, Maslov-corrected — and the failure of the scheme in chaotic regions, exactly as Einstein foresaw, is a founding observation of quantum chaos. In accelerator physics, beam emittances are transverse actions and their conservation budgets the luminosity of every collider; in celestial mechanics, the proper elements used to classify asteroid families are actions computed to higher order. Wherever slow, bounded, nearly-regular motion must be labelled by robust numbers, the labels in use are Liouville's loop integrals.

The physical picture

Picture a doughnut floating in phase space. There are two fundamentally different ways to wrap a rubber band around it: the short way around the tube, and the long way around the central hole. No amount of sliding, stretching, or wiggling converts one wrap into the other — they are distinct homology classes, and the doughnut has exactly two. An $n$ -torus likewise has exactly $n$ . The action $J_i$ is the circulation of canonical momentum captured by the $i$ -th rubber band; and because the one-form being integrated is closed on the torus, you may slide the band anywhere you like without changing the number. The actions are not properties of a path. They are properties of the doughnut.

Figure

The invariant torus and its two irreducible cycles — slide a loop freely, its action integral does not change

Where do the doughnuts come from? From conservation laws. Each conserved quantity in involution confines the motion to one level set; $n$ of them, independent and commuting, cut the $2n$ -dimensional phase space down to an $n$ -dimensional surface that the trajectory can never leave. The Liouville-Arnold Theorem (its own article in this chapter) supplies the deep structural facts: when that surface is compact and connected it must be a torus — commuting flows rule out spheres and everything else — and the flow on it is conditionally periodic. This article takes the tori as given and asks what natural numbers live on them. For one degree of freedom you already know the answer in disguise. The "torus" $T^1$ is just the closed energy contour in the $(q,p)$ plane, and $2\pi J$ is the area it encloses — phase-space area, the same currency in which Liouville's theorem measures volume and quantum mechanics counts states ( $J$ has the units of $\hbar$ ; no accident). The pendulum shows why topology, not just area, matters: below the separatrix energy the contour is a closed lens (a libration — the band wraps the tube), while above it the contour winds around the angular coordinate without ever turning back (a rotation — the band wraps the hole of the cylindrical phase space). Two homology classes, two inequivalent actions, at any energy you like.

Figure

Libration and rotation on the pendulum's phase cylinder — two homology classes of loops, two kinds of action

One warning before the mathematics, because it trips everyone once: the cycles $\gamma_i$ are not what the system does in time. A trajectory with incommensurate frequencies covers its torus densely and never closes. The cycles are surveyor's loops — drawn on the frozen torus at fixed values of the conserved quantities, one for each independent way of going around. You compute actions from the geometry of the invariant surface, never by following the motion.

The action integrals

Fix an integrable system: $n$ conserved quantities $F_1,\dots,F_n$ (one of them the Hamiltonian), independent and in involution, $\{F_i,F_j\}=0$ . Freeze their values; the resulting level set — compact and connected — is an invariant torus, and on it the momenta become (locally) functions $p_k(q;f)$ of the coordinates. The central definition integrates the canonical one-form around each fundamental cycle:

J_i=\frac{1}{2\pi}\oint_{\gamma_i}\mathbf{p}\cdot\mathrm{d}\mathbf{q}=\frac{1}{2\pi}\oint_{\gamma_i}\sum_{k=1}^{n}p_k\,\mathrm{d}q_k \qquad (1)

where: $\gamma_i$ — the $i$ -th irreducible cycle of the torus; $J_i$ — the $i$ -th action, with units of energy × time (the units of $\hbar$ ). Read the structure before the details: the integrand is a fixed geometric object (the canonical one-form $\lambda=\sum_kp_k\,\mathrm{d}q_k$ , the same one whose exterior derivative is the symplectic form), and the only choice made is which class of loop to wrap. The $1/2\pi$ is a normalization with a job — it is what will make the conjugate variables honest $2\pi$ -periodic angles, a bookkeeping handled in Action-Angle Variables. Why exactly $n$ integrals, no more, no fewer? Because that is how many independent loops a torus owns:

H_1(T^n)\cong\mathbb{Z}^n \qquad (2)

where: $H_1$ — the first homology group, whose elements are loops-up-to-deformation with integer winding labels. A loop on $T^n$ is classified by $n$ integers (how many times it winds around each fundamental direction); the $n$ generators give $n$ independent actions, and any other loop's integral is the corresponding integer combination of them. Topology fixes the count of actions the way dimension fixes the count of coordinates. In a separable system the abstract cycles become concrete: $\gamma_i$ is the loop along which $q_i$ runs through one full cycle of its own one-dimensional motion — out and back between turning points for a libration, once around for a rotation — while every other coordinate sits still. Then only the $k=i$ term of (1) survives, and each action collapses to a single-coordinate integral,

J_i=\frac{1}{2\pi}\oint p_i(q_i;\alpha)\,\mathrm{d}q_i \qquad (3)

where: $p_i(q_i;\alpha)$ — the separated momentum delivered by the machinery of Separation of Variables in Hamilton-Jacobi Theory, with the separation constants $\alpha$ as frozen parameters. This is the form in which actions are actually computed, and it makes the constancy of $J_i$ manifest: the integral depends only on the constants $\alpha$ , so it is itself a constant — the separation constants repackaged into their geometrically canonical form. The actions connect back to Hamilton-Jacobi theory through the characteristic function. Along the torus, Hamilton's Characteristic Function is the line integral of the very same one-form,

W(q;J)=\int^{q}\mathbf{p}(q';J)\cdot\mathrm{d}\mathbf{q}',\qquad p_k=\frac{\partial W}{\partial q_k} \qquad (4)

where: the integral runs from a reference point to $q$ on the torus. But the one-form has nonzero loop integrals — so $W$ cannot be single-valued. Carry $W$ once around the cycle $\gamma_i$ and it comes back shifted:

\Delta_iW\equiv\oint_{\gamma_i}\mathrm{d}W=\oint_{\gamma_i}\sum_k\frac{\partial W}{\partial q_k}\,\mathrm{d}q_k=\oint_{\gamma_i}\sum_kp_k\,\mathrm{d}q_k=2\pi J_i \qquad (5)

where: the middle equality uses $p_k=\partial W/\partial q_k$ . The multivaluedness is not a defect; it is the point. The periods of $W$ are exactly the actions, and this is the property that makes $W(q,J)$ — used as a type-2 generating function — produce conjugate variables that advance by precisely $2\pi$ per cycle. That construction, and everything that follows from it (the angle variables, $H=H(J)$ , and the frequencies $\omega_i=\partial H/\partial J_i$ ), is the business of the next two articles; here it is enough to see why the actions, of all possible relabelings of the conserved quantities, are the canonical choice.

Which equation answers which question

you want	use	notes
the general definition	eq. (1)	any integrable system, any cycle basis
how many independent actions	eq. (2)	exactly $n$ — topology decides
to actually compute $J_i$	eq. (3)	separable case: one-coordinate loop integrals
the 1-DOF shortcut	$2\pi J=$ enclosed $(q,p)$ area	libration: lens area; rotation: one strip
why loops may be deformed	Derivation B	the torus is Lagrangian; the one-form is closed
what the actions are for	eqs. (4)–(5)	periods of $W$ — seed of action-angle variables

Derivation AWhy the invariant torus is a Lagrangian surface

We prove the geometric fact on which this whole article rests: the symplectic form $\omega=\sum_k\mathrm{d}p_k\wedge\mathrm{d}q_k$ restricted to an invariant torus vanishes identically — the torus is a Lagrangian submanifold. The strategy is to build a basis of the torus's tangent space out of the Hamiltonian vector fields of the conserved quantities and then evaluate $\omega$ on that basis, where every entry becomes a Poisson bracket. The involution hypothesis then kills everything. Watch how each hypothesis of the Liouville-Arnold setup gets used exactly once: independence gives a basis, invariance keeps the fields tangent, involution zeroes the brackets.

Step 1. For each conserved quantity $F_i$ , let $X_i$ be its Hamiltonian vector field — the phase-space flow that $F_i$ would generate as a Hamiltonian. Each $X_i$ is tangent to the level set $M_f=\{F_j=f_j\}$ . Why: the flow of $X_i$ preserves every $F_j$ , because the rate of change of $F_j$ along it is $\{F_j,F_i\}=0$ by involution — so the flow cannot leave the common level set.

Step 2. The $n$ fields $X_1,\dots,X_n$ form a basis of the tangent space of $M_f$ at every point. Why: independence of the $F_i$ means their gradients are linearly independent, and the map from gradients to Hamiltonian fields is an isomorphism; $n$ independent tangent fields on an $n$ -dimensional surface are a basis.

Step 3. Evaluate the symplectic form on the basis. The defining property of Hamiltonian vector fields converts symplectic products into Poisson brackets:

\omega(X_i,X_j)=\{F_i,F_j\}=0

Why this is the crux: the two-form $\omega$ is fully determined on $M_f$ by its values on a tangent basis, and involution makes every such value vanish.

Step 4. Conclude $\omega\big|_{M_f}=0$ : the invariant torus is Lagrangian — an $n$ -dimensional surface in $2n$ -dimensional phase space on which the symplectic area of every infinitesimal parallelogram is zero. Why $n$ is the maximum possible: a submanifold on which $\omega$ vanishes can have dimension at most $n$ (half the phase space), so invariant tori are as large as Lagrangian surfaces get — "maximally flat" carriers of the dynamics. *Verified: this is Arnold §49, Lemma-level content; the same computation shows conversely that the tangent flows commute, which is the seed of the Liouville-Arnold theorem's identification of $M_f$ with a torus.*

Go deeper · Derivation B · 2 minContour independence — the action belongs to the homology class

Derivation BContour independence — the action belongs to the homology class

We prove the robustness property announced in the overview: two loops on the torus that can be deformed into each other give the same action integral, so $J_i$ is a function of the homology class $[\gamma_i]$ alone — a label of the torus, not of any path. The strategy is two lines of exterior calculus sitting on Derivation A: the canonical one-form is closed on the torus because its derivative is the symplectic form, which vanishes there; Stokes' theorem then converts the difference of two loop integrals into the integral of that vanishing derivative. The subtlety worth watching: closedness holds only on the torus — in the full phase space $\mathrm{d}\lambda=\omega\neq0$ — so loops must be deformed within the invariant surface.

Step 1. Restrict the canonical one-form $\lambda=\sum_kp_k\,\mathrm{d}q_k$ to the invariant torus and take its exterior derivative:

\mathrm{d}\lambda\big|_{T^n}=\sum_k\mathrm{d}p_k\wedge\mathrm{d}q_k\Big|_{T^n}=\omega\big|_{T^n}=0

Why: the exterior derivative commutes with restriction, and Derivation A established that the symplectic form vanishes on the torus. On the torus — and only there — $\lambda$ is a closed form.

Step 2. Let $\gamma$ and $\gamma'$ be homologous cycles: together (one traversed backwards) they bound a two-dimensional strip $\Sigma$ lying inside the torus. Apply Stokes' theorem on $\Sigma$ :

\oint_{\gamma}\lambda-\oint_{\gamma'}\lambda=\int_{\Sigma}\mathrm{d}\lambda=0

Why the orientation bookkeeping is safe: "homologous" means precisely that the difference cycle is a boundary within the surface; Stokes converts the boundary integral into the surface integral of the (vanishing) two-form.

Step 3. Conclude $J_i=\frac{1}{2\pi}\oint_{\gamma_i}\lambda$ is unchanged under any continuous deformation of $\gamma_i$ within the torus: the action is a homology invariant. Non-homologous loops — the pendulum's libration lens versus its rotation strip — bound no common strip, and their integrals are genuinely different numbers.

Step 4. Draw the physical consequence. Since any convenient representative computes the same $J_i$ , one may choose the separable representative (only $q_i$ moving) and reduce the action to the one-dimensional integral (3) — the deformation freedom is what licenses that reduction even when the physically natural loops are tilted or wiggly. Verified: Arnold §49 — the canonical one-form restricted to an invariant Lagrangian torus is closed, and its loop integrals define the actions on homology classes; consistent with the explicit pendulum computation in Derivation C.

Go deeper · Derivation C · 2 minThe pendulum's two actions — libration and rotation loops evaluated

Derivation CThe pendulum's two actions — libration and rotation loops evaluated

We compute the action integrals of the plane pendulum, $H=\dfrac{p^2}{2ml^2}-mgl\cos\theta$ , in both homology classes — the libration lens below the separatrix and the rotation strip above it — and extract the exact separatrix limits, where the two classes meet with a famous factor of two. The strategy: write $p(\theta)$ from energy conservation, convert each loop integral to a complete elliptic integral by the half-angle substitution, and take limits. Track the moduli carefully: the libration modulus $k=\sin(\theta_0/2)$ is built from the amplitude, the rotation modulus $\tilde k$ from the energy — they tend to 1 from opposite sides of the separatrix.

Step 1. Libration setup. For $-mgl<E<mgl$ the pendulum swings between turning points $\pm\theta_0$ with $E=-mgl\cos\theta_0$ . Energy conservation gives, with $\omega_0=\sqrt{g/l}$ and the half-angle identity $\cos\theta=1-2\sin^2(\theta/2)$ :

p(\theta)=\sqrt{2ml^2\big(E+mgl\cos\theta\big)}=2ml^2\omega_0\sqrt{k^2-\sin^2\tfrac{\theta}{2}},\qquad k\equiv\sin\tfrac{\theta_0}{2}

Why the half-angle form: it turns the turning-point condition into $\sin(\theta/2)=k$ — the standard gateway to elliptic integrals.

Step 2. Libration action. The closed lens is traversed once: $\oint=4\int_0^{\theta_0}$ , so with $\theta=2x$ and then $\sin x=k\sin\varphi$ :

J_{\mathrm{lib}}=\frac{4}{2\pi}\int_0^{\theta_0}p\,\mathrm{d}\theta=\frac{8ml^2\omega_0}{\pi}\int_0^{\arcsin k}\!\sqrt{k^2-\sin^2x}\;\mathrm{d}x=\frac{8ml^2\omega_0}{\pi}\Big[E(k)-(1-k^2)K(k)\Big]

where: $K$ and $E$ — complete elliptic integrals of the first and second kind. Why the bracket: the substitution $\sin x=k\sin\varphi$ maps the integral onto $k^2\int_0^{\pi/2}\cos^2\varphi\,(1-k^2\sin^2\varphi)^{-1/2}\mathrm{d}\varphi$ , a standard reduction equal to $E(k)-(1-k^2)K(k)$ .

Step 3. Rotation setup and action. For $E>mgl$ the momentum never vanishes; write $E+mgl\cos\theta=(E+mgl)\big(1-\tilde k^2\sin^2\tfrac{\theta}{2}\big)$ with $\tilde k^2\equiv\dfrac{2mgl}{E+mgl}<1$ , so $p=\dfrac{2ml^2\omega_0}{\tilde k}\sqrt{1-\tilde k^2\sin^2\tfrac{\theta}{2}}$ . The rotation cycle crosses $\theta$ once from $0$ to $2\pi$ (no doubling back):

J_{\mathrm{rot}}=\frac{1}{2\pi}\int_0^{2\pi}p\,\mathrm{d}\theta=\frac{2ml^2\omega_0}{\pi\tilde k}\int_0^{\pi}\sqrt{1-\tilde k^2\sin^2x}\;\mathrm{d}x\cdot\frac{2}{2}=\frac{4ml^2\omega_0}{\pi\tilde k}\,E(\tilde k)

Why no factor of 4: the rotation loop wraps the cylinder once with $p$ of a single sign — the homology class, not habit, dictates the integration range.

Step 4. Separatrix limits. As $E\to mgl$ from below, $k\to1$ : using $E(1)=1$ and $(1-k^2)K(k)\to0$ , $J_{\mathrm{lib}}\to\dfrac{8ml^2\omega_0}{\pi}$ . From above, $\tilde k\to1$ : $J_{\mathrm{rot}}\to\dfrac{4ml^2\omega_0}{\pi}$ — exactly half the libration limit.

Step 5. Read the factor of two topologically. At the separatrix the libration lens degenerates into the union of the upper and lower separatrix branches — which is precisely the rotation loop plus its momentum-reversed mirror image. One class limits onto two copies of the other, and the actions say so: $J_{\mathrm{lib}}\to2\,J_{\mathrm{rot}}$ . Comparing actions across a separatrix without tracking homology is meaningless — the classes, not just the numbers, change. Verified: both elliptic-integral forms match Landau-Lifshitz §49 problem material and standard references; limit check $k\to0$ : $E(k)-(1-k^2)K(k)\to\pi k^2/4$ , giving $J_{\mathrm{lib}}\to\tfrac12 ml^2\omega_0\theta_0^2=E_{\mathrm{osc}}/\omega_0$ , the small-oscillation result.

Limiting cases

Case	Result	Meaning
one degree of freedom	$2\pi J=$ area enclosed in the $(q,p)$ plane	every bounded 1-DOF system is integrable; the torus is the energy contour itself
harmonic oscillator	$J=E/\omega$	the ellipse area $2\pi E/\omega$ over $2\pi$ ; linear $H(J)$ is the fingerprint of isochronism
separable system	$J_i=\frac{1}{2\pi}\oint p_i(q_i;\alpha)\,\mathrm{d}q_i$	each cycle lives in its own $(q_i,p_i)$ plane; actions = repackaged separation constants
approach to a separatrix	$J$ finite, but $\mathrm{d}J/\mathrm{d}E\to\infty$	the torus family degenerates; libration and rotation classes collide with $J_{\mathrm{lib}}\to2J_{\mathrm{rot}}$
commensurate frequencies	actions not unique — integer recombinations allowed	degenerate tori admit different cycle bases; the Kepler problem is the showcase (see Degeneracy and Commensurate Frequencies)
non-integrable dynamics	no invariant tori ⇒ no actions	in chaotic regions the loop integrals lose meaning entirely — actions are creatures of integrability

Common trapThe action cycle is the loop the system traces in time

It is not — and this single confusion generates more wrong action integrals than any other. A generic trajectory on a torus with incommensurate frequencies winds densely and never closes: following the motion for one "period of $q_1$ " does not return you to your starting phase-space point, and the line integral along actual motion is not $2\pi J_1$ . The cycle $\gamma_i$ is a surveyor's loop drawn on the frozen geometry of the torus — all conserved quantities held fixed, only $q_i$ taken around its circuit, the other coordinates returned to base by construction, not by dynamics. The license to compute this way comes from Derivation B: the one-form is closed on the torus, so only the homology class matters, and the convenient separable representative is as good as any physical-looking loop. Compute actions from geometry; reserve trajectories for checking frequencies afterwards.

Worked examples

1 · Phase-ellipse bookkeeping for a laboratory oscillator (confidence builder)

A mass $m=0.2\ \mathrm{kg}$ on a spring oscillates at $\omega=5.0\ \mathrm{rad/s}$ with amplitude $A=0.10\ \mathrm{m}$ . Compute the action $J$ three ways: from the enclosed phase-plane area, from $J=E/\omega$ , and check the units.

Step 1. The orbit in the $(q,p)$ plane is the ellipse $\frac{p^2}{2mE}+\frac{q^2}{2E/m\omega^2}=1$ with semi-axes $q_{\max}=A=0.10\ \mathrm{m}$ and $p_{\max}=m\omega A=0.2\times5.0\times0.10=\mathbf{0.10\ kg\,m/s}$ .

Step 2. Area route: the ellipse area is $\pi q_{\max}p_{\max}=\pi\times0.10\times0.10=0.0314\ \mathrm{J\,s}$ , so $J=\dfrac{\text{area}}{2\pi}=\mathbf{5.0\times10^{-3}\ J\,s}$ .

Step 3. Energy route: $E=\tfrac12 m\omega^2A^2=\tfrac12\times0.2\times25\times0.01=0.025\ \mathrm{J}$ , hence $J=E/\omega=0.025/5.0=\mathbf{5.0\times10^{-3}\ J\,s}$ — the two routes agree, as Derivation B's contour freedom guarantees they must.

Step 4. Units and scale: $J$ carries $\mathrm{J\,s}$ , the units of $\hbar=1.055\times10^{-34}\ \mathrm{J\,s}$ . This single laboratory oscillation encloses $J/\hbar\approx4.7\times10^{31}$ quanta of phase area — the measure of how deep in the classical regime a tabletop spring lives.

2 · The pendulum's libration action, exactly and approximately (exam level)

A pendulum with $m=1.0\ \mathrm{kg}$ , $l=1.0\ \mathrm{m}$ ( $g=9.81\ \mathrm{m/s^2}$ ) librates with amplitude $\theta_0=90°$ . Evaluate its action exactly, compare with the harmonic approximation, and locate the energy relative to the separatrix.

Step 1. Parameters: $\omega_0=\sqrt{g/l}=3.132\ \mathrm{rad/s}$ ; modulus $k=\sin(\theta_0/2)=\sin45°=0.7071$ , so $k^2=0.500$ .

Step 2. Elliptic integrals at $k^2=0.5$ : $K(k)=1.8541$ , $E(k)=1.3506$ . The bracket of Derivation C: $E(k)-(1-k^2)K(k)=1.3506-0.5\times1.8541=0.4236$ .

Step 3. Exact action: $J_{\mathrm{lib}}=\dfrac{8ml^2\omega_0}{\pi}\times0.4236=\dfrac{8\times1\times1\times3.132}{\pi}\times0.4236=7.976\times0.4236=\mathbf{3.38\ J\,s}$ .

Step 4. Harmonic estimate: energy above the bottom of the well is $E_{\mathrm{osc}}=mgl(1-\cos\theta_0)=9.81\times(1-0)=9.81\ \mathrm{J}$ , giving $J_{\mathrm{harm}}=E_{\mathrm{osc}}/\omega_0=9.81/3.132=\mathbf{3.13\ J\,s}$ . The exact action exceeds it by $8\%$ : the softening potential fattens the phase-plane lens relative to the harmonic ellipse of equal energy — anharmonicity, measured in phase area.

Step 5. Position relative to the separatrix: the separatrix action is $J_{\mathrm{sep}}=8ml^2\omega_0/\pi=7.98\ \mathrm{J\,s}$ , so this orbit carries $3.38/7.98=42\%$ of the maximal libration action, comfortably below the danger zone. (And a rotation orbit at the separatrix would carry $J_{\mathrm{rot}}=3.99\ \mathrm{J\,s}$ — the factor-of-two bookkeeping of Derivation C, Step 5, on live numbers.)

The geometer behind the tori

Origins · 1 minJoseph Liouville and how this was discovered

Joseph Liouville

1809–1882 · Collège de France, Paris

Liouville is present twice in this subject, under two theorems. His phase-volume theorem gave mechanics its incompressible-flow picture of phase space; but the result that owns this chapter is his 1855 observation that a Hamiltonian system with $n$ independent commuting integrals can be solved by quadratures — no further integration of differential equations required. Liouville worked analytically; the geometry hiding in his theorem — that the joint level sets are tori, that the flow on them is straight-line winding, that the quadratures are loop integrals over homology cycles — waited more than a century for Vladimir Arnold to make explicit, which is why the modern statement carries both names. Liouville's own career was a study in range: transcendental numbers, Sturm-Liouville theory, differential geometry, and the editorship of the Journal de Mathématiques, where he rescued Galois's manuscripts from oblivion.

History

1855

— Liouville: n commuting independent integrals reduce a Hamiltonian system to quadratures — integrability acquires its definition.

1889

— Poincaré's prize memoir on the three-body problem shows generic systems lack such integrals: the tori are special, not universal.

1917

— Einstein reformulates the Bohr-Sommerfeld rules as integrals over the homology cycles of invariant tori, flagging that chaotic motion admits no such quantization.

1954

— Kolmogorov announces that strongly irrational tori survive small perturbations — the opening move of KAM theory, with actions as the surviving labels.

1963

— Arnold publishes the modern proof that compact connected level sets of commuting integrals are tori with conditionally periodic flow: the Liouville-Arnold theorem.

Going further

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §10.6–10.8 · Landau & Lifshitz, Mechanics (Vol 1), §50 · Arnold, Mathematical Methods of Classical Mechanics, §49–50 · Taylor, Classical Mechanics, Ch. 13 · Morin, Introduction to Classical Mechanics, Ch. 15 (Hamiltonian methods)

Provenance. Definitions and the SHO result $J=E/\omega$ follow Goldstein §10.6–10.8 and Landau–Lifshitz §49–50; the closedness/Lagrangian-torus argument and $H_1(T^n)\cong\mathbb{Z}^n$ follow Arnold §49. A verifier should recompute: the ellipse area $2\pi E/\omega$ giving $J=E/\omega$ and $\omega=\partial H/\partial J$ ; the increment $\oint_{\gamma_i}\mathrm{d}W=2\pi J_i$ and $\oint_{\gamma_j}\mathrm{d}\theta_i=2\pi\delta_{ij}$ ; the hydrogen sum $n=n_r+n_\theta+n_\phi$ with $E_n=-mZ^2e^4/(2\hbar^2n^2)$ ; and the first-order cancellation in the adiabatic derivation. All numbers are standard textbook values. Worked-example and derivation numbers on this page ( $J=5.0\times10^{-3}$ J s for the spring; $k^2=0.5$ , $K=1.8541$ , $E=1.3506$ , $J_{\mathrm{lib}}=3.38$ J s, $J_{\mathrm{sep}}=7.98$ J s for the pendulum; $J_{\mathrm{lib}}\to2J_{\mathrm{rot}}$ at the separatrix) are illustrative arithmetic recomputed from the displayed formulas.

Builds on: Liouville-Arnold Theorem · Liouville Integrable Systems · Hamilton's Characteristic Function · Separation of Variables in Hamilton-Jacobi Theory Leads to: Action-Angle Variables · Frequencies from Action-Angle Variables · Degeneracy and Commensurate Frequencies

Why put the 1/2π in the definition instead of using the bare loop integral?

Both conventions exist in the literature, and the difference is pure bookkeeping — but bookkeeping with consequences. With the $1/2\pi$ (Goldstein's and Arnold's modern convention), the variable conjugate to $J_i$ advances by exactly $2\pi$ per cycle: it is an honest angle, and its rate is an honest angular frequency in rad/s. With the bare integral $\bar J=\oint p\,\mathrm{d}q=2\pi J$ (Landau's convention, and old quantum theory's $\oint p\,\mathrm{d}q=nh$ ), the conjugate variable ticks by 1 per cycle and its rate is a frequency in cycles per second. Every factor-of- $2\pi$ error in this subject is a collision between the two conventions. Check which one your source uses before importing any formula — the tell is whether the harmonic oscillator comes out as $J=E/\omega$ or $\bar J=2\pi E/\omega=E/\nu$ .

Are the actions of a given torus unique, or is there freedom in choosing them?

There is precisely the freedom of choosing a basis of loops. The homology group $\mathbb{Z}^n$ admits different generator sets, related by integer matrices with determinant $\pm1$ (unimodular transformations); the corresponding actions mix as $J_i'=\sum_jM_{ij}J_j$ with the same integer matrices. For a nondegenerate torus the physically natural basis — separable cycles, one per coordinate — is essentially canonical, and everyone computes the same $J_i$ . Degeneracy changes the story: when frequencies are commensurate, genuinely different bases are equally natural (spherical versus parabolic cycles for Kepler), and different action sets coexist. That non-uniqueness is not a defect; it is the geometric signature of degeneracy, taken up in Degeneracy and Commensurate Frequencies.

Is this J the same "action" as in the principle of least action?

Same currency, different objects. Hamilton's principle extremizes $\int L\,\mathrm{d}t$ along timed paths between fixed endpoints; the abbreviated (Maupertuis) action $\int\mathbf{p}\cdot\mathrm{d}\mathbf{q}$ drops the schedule and is extremized along geometric paths at fixed energy. The action integral $J$ is the abbreviated action evaluated around a closed topological cycle of the invariant torus — no endpoints, no extremization, one number per homology class. The shared integrand is why all three carry units of $\hbar$ and why the name is overloaded. When a text says "the action is quantized" (Bohr-Sommerfeld) it always means $J$ ; when it says "the action is stationary" it means one of the other two. The article on Hamilton's Principal Function untangles the family tree.

How can the one-form be closed on the torus when dλ = ω is the symplectic form, which is famously non-degenerate?

Because restriction is a brutal operation. In the full $2n$ -dimensional phase space $\mathrm{d}\lambda=\omega$ is indeed nowhere-vanishing — that non-degeneracy is what makes phase space symplectic. But evaluating a two-form requires feeding it two tangent vectors, and on the invariant torus every tangent vector is a combination of the commuting Hamiltonian fields $X_i$ of the conserved quantities. All their pairings are Poisson brackets $\{F_i,F_j\}=0$ (Derivation A), so the restriction $\omega|_{T^n}$ vanishes identically even though $\omega$ itself does not. "Lagrangian submanifold" is exactly this situation's name: a half-dimensional surface threading phase space along directions the symplectic form cannot see. Closedness of $\lambda$ — and with it the whole contour freedom of the actions — lives on the torus and nowhere else.

What actually happens to the action of an orbit as it crosses a separatrix?

Strictly, an integrable system's orbit never crosses — the separatrix is itself an invariant set, reached only asymptotically. The question bites when something slow (a parameter drift, a perturbation) carries orbits across. Then three things happen. First, the homology class changes discontinuously: a libration lens is not deformable into a rotation strip, and the bookkeeping $J_{\mathrm{lib}}\to2J_{\mathrm{rot}}$ at the crossing is forced by the lens degenerating into the strip plus its mirror image. Second, the adiabatic protection of the action fails right at the crossing — the period diverges there, so no drift is "slow" compared to it — and orbits pick up a quasi-random action jump of order the drift rate times a logarithm. Third, an ensemble of orbits is split probabilistically between the two rotation senses. Separatrix crossing theory quantifies all three and is a working tool in plasma physics and celestial mechanics.

Do the actions exist for scattering (unbounded) motion?

The torus story needs compactness: the Liouville-Arnold theorem's identification of level sets with $T^n$ holds when the joint level set is compact and connected. Unbounded motion — a hyperbolic Kepler orbit, a particle rolling over a potential hill — lives on level sets diffeomorphic to cylinders $\mathbb{R}^k\times T^{n-k}$ , which have fewer independent cycles: only the compact directions contribute loop integrals. You can still define actions for the periodic pieces (the angular action of a scattering orbit survives; the radial one does not), and the non-compact directions are handled by other normal forms. The clean slogan: actions count the closed directions of the invariant geometry, and only bounded integrable motion has all $n$ of them.

Why are actions singled out as "the" labels of tori when any function of the conserved quantities would also be constant?

Three properties conspire, and only the actions have all three. They are canonical: unlike a random relabelling $g(F_1,\dots,F_n)$ , the actions extend to half of a canonical coordinate pair — their conjugates tick uniformly by $2\pi$ per cycle, which is what makes $H=H(J)$ solve the dynamics outright (the construction in Action-Angle Variables). They are geometric: defined through homology, they survive any change of coordinates and any choice of which $n$ commuting functions you happened to conserve. And they are physical: the same loop integrals are the adiabatic invariants — the quantities protected under slow change (a headline result of the Action-Angle Variables article) — and the quantities the semiclassical rules quantize in units of $\hbar$ . Energy tells you how expensive a torus is; the actions tell you which torus it is.

Is 2πJ literally an area also when n is bigger than 1?

Per cycle, yes — as a sum of projected areas. Choose the separable representative of $\gamma_i$ : only $q_i$ moves, so $2\pi J_i=\oint p_i\,\mathrm{d}q_i$ is the area of the loop that the pair $(q_i,p_i)$ traces in its own phase plane, other coordinates frozen. For a general (tilted, wiggly) representative the integral picks up contributions $\oint p_k\,\mathrm{d}q_k$ from every pair — each the signed area of that pair's projection — and closedness guarantees the sum is deformation-independent even though the individual projections are not. What is not true is any claim about $2$ -dimensional area on the torus itself: the symplectic area of the torus is zero (it is Lagrangian!), which is exactly why the one-dimensional loop integrals, rather than any surface integrals, carry the invariant information.