Frequencies from Action-Angle Variables

Ask a classical mechanic the most expensive question in the subject — "how fast does it oscillate?" — and the naive route is brutal: solve the equations of motion, wait for the trajectory to repeat, read off the period. Action-angle theory replaces all of that with a derivative. For any integrable system, once the energy is known as a function of the action variables, every fundamental frequency of the motion is a slope: $\omega_i=\partial H/\partial J_i$ . No trajectory is ever computed. The recipe runs entirely through quadratures: evaluate the action integrals $J_i(E,\alpha)$ over the invariant tori, invert to get $H(J)$ , differentiate. The pendulum's amplitude-dependent period, the quartic oscillator's hardening spectrum, the precession rate of a perturbed planetary ellipse — each is one differentiation away from an integral table. This article is about wielding that formula. The construction that justifies it — the canonical transformation to action-angle variables, built from Hamilton's characteristic function — is the business of the Action-Angle Variables article; the geometric meaning of the actions themselves lives in Action Integrals and Invariant Tori. Here we take the machine as built and extract its output: the practical one-degree-of-freedom identity (the period is the rate at which enclosed phase-space area grows with energy), the multi-frequency anatomy of tori, and the diagnostic power of frequency ratios — equal frequencies expose the hidden degeneracy of the Kepler problem, and frequency splittings measure the precession of nearly closed orbits. When the ratios are rational, orbits close; when they drift with amplitude, chaos theory takes notice: the frequency map $\omega(J)$ is the object on which all of perturbation theory stands.

Key takeaways

In action-angle variables the Hamiltonian depends on the actions alone, $H=H(J)$ , so Hamilton's equations collapse: actions frozen, angles advancing uniformly.
The fundamental result is $\omega_i=\partial H/\partial J_i$ — every frequency of a multiply periodic motion is a derivative of the energy with respect to an action, no orbit integration required.
The working recipe: compute $J_i(E)$ by quadrature, then differentiate; for one degree of freedom, $T=2\pi\,\partial J/\partial E$ — the period is the growth rate of enclosed phase-space area with energy, and no explicit inversion to $H(J)$ is ever needed.
Linear $H(J)$ means amplitude-independent frequency (the harmonic oscillator); curvature of $H(J)$ is anharmonicity, $\mathrm{d}\omega/\mathrm{d}J\neq0$ .
Frequency ratios diagnose the dynamics: all-equal frequencies signal the complete degeneracy of Kepler's problem (closed ellipses); a small splitting $\omega_\phi\neq\omega_r$ is the apsidal precession rate of a perturbed orbit.
Key idea: in action-angle coordinates $H=H(J)$ alone, so $\dot\theta_i=\partial H/\partial J_i\equiv\omega_i$ is constant — the frequencies of a multiply periodic system are obtained by differentiating the energy with respect to the actions, no orbit integration required.

Where this lives today

Every circular accelerator on Earth is operated through this article's formula. The transverse oscillations of a stored beam are characterized by their tunes — the betatron frequencies per revolution, which are exactly $\partial H/\partial J$ for the transverse actions — and the "working point" of a collider is a chosen pair of tunes placed, with surgical care, away from every low-order resonance line $n_1\omega_1+n_2\omega_2=$ integer. Tune measurement, tune feedback, and resonance compensation are daily operations at the LHC and at every light source; when machine physicists map beam loss against tune space, they are drawing equation (5) on an oscilloscope. The same frequency-first viewpoint powers modern dynamical astronomy and gravitational-wave science. Laskar's frequency-map analysis extracts the fundamental frequencies $\omega_i(J)$ numerically from short orbit segments and watches them drift — a drift of frequency being the sharpest available diagnostic of chaos, applied from asteroid belts to exoplanet systems. And the extreme mass-ratio inspirals that LISA is being designed to hear are modelled as slowly evolving Kerr geodesics, catalogued by their three fundamental frequencies $(\omega_r,\omega_\theta,\omega_\phi)$ ; the gravitational waveform's spectrum is the integer lattice $n_1\omega_1+n_2\omega_2+n_3\omega_3$ of Derivation-B-style splittings, chirping as radiation reaction walks the orbit through action space.

The physical picture

Plot the energy of an integrable system against its action. That single curve $H(J)$ carries the entire kinematics of the motion: its slope at your orbit's action is your orbit's angular frequency. A harmonic oscillator draws a straight line, $H=\omega_0J$ — same slope everywhere, which is why its frequency famously ignores amplitude. Any real, anharmonic system draws a curve, and the bending is physical: frequency that changes with amplitude. In this picture "nonlinearity" stops being a vague adjective and becomes a second derivative, $\mathrm{d}\omega/\mathrm{d}J=\partial^2H/\partial J^2$ — the number that KAM theory will later demand be nonzero.

Figure

The energy-action curve is the whole story — its slope is the frequency, its curvature is the anharmonicity

For one degree of freedom the formula has a beautifully concrete geometric reading. The action is the enclosed phase-plane area over $2\pi$ (Action Integrals and Invariant Tori); its derivative with respect to energy asks: when I raise the energy by $\mathrm{d}E$ , how much annular area opens up between the old orbit and the new? The answer — proved in Derivation A — is that the annulus area grows at exactly the rate $T$ , the orbital period. Slow orbits are slow precisely where energy contours crowd apart; fast orbits live where contours pack tight. The frequency formula is contour cartography.

Figure

Fatten the orbit by dE and the annular area grows at rate T — the period as an area derivative

With several degrees of freedom the picture is the torus of Action Integrals and Invariant Tori, now with clocks attached. Each independent cycle of the torus carries its own uniformly ticking angle, and $\omega_i=\partial H/\partial J_i$ sets the $n$ rates. Everything observable about the motion is woven from these rates: the coordinates are multiply periodic, so any $q(t)$ is a Fourier series over integer combinations $n_1\omega_1+\dots+n_n\omega_n$ . Whether the trajectory closes (rational frequency ratios) or carpets its torus densely (irrational) is decided by arithmetic — and when a perturbation splits two formerly equal frequencies, the orbit visibly precesses at exactly the splitting. That last effect, computed in Derivation B, is the classical mechanic's stethoscope: planetary apsides, satellite orbits and accelerator beams are all monitored through their frequency splittings.

The frequency machinery

The starting point is the endpoint of the Action-Angle Variables construction: a canonical transformation $(q,p)\to(\theta,J)$ , generated by Hamilton's characteristic function, in which the new Hamiltonian carries no angle dependence,

H=H(J_1,\dots,J_n) \qquad (1)

where: $J_i$ — the action variables, loop integrals $\frac{1}{2\pi}\oint p_i\,\mathrm{d}q_i$ over the torus cycles; the absence of the $\theta_i$ is the defining achievement of the transformation, not an assumption. Every angle is cyclic: the system is "solved" in the same sense that a free particle is solved. Hamilton's equations then deliver the dynamics in two lines. The angle equation is the harvest:

\dot J_i=-\frac{\partial H}{\partial\theta_i}=0,\qquad \dot\theta_i=\frac{\partial H}{\partial J_i}\equiv\omega_i(J) \qquad (2)

where: the first equation confirms the actions are frozen; the second defines the frequencies. Because each $\omega_i$ is a function of the frozen actions only, it is a constant of the motion — evaluated once, valid forever. Integrating the constant rate:

\theta_i(t)=\omega_i\,t+\theta_i(0),\qquad T_i=\frac{2\pi}{\omega_i} \qquad (3)

where: $\theta_i(0)$ — the starting phase on the torus; $T_i$ — the period of the $i$ -th angle. This is the cleanest statement multiply periodic motion admits: uniform straight-line winding, with all the difficulty of the original nonlinear problem compressed into computing the function $H(J)$ . For one degree of freedom the formula acquires its most-used practical form. There is a single action $J(E)$ , and rather than inverting it to $H(J)$ one differentiates implicitly:

\omega=\left(\frac{\partial J}{\partial E}\right)^{-1},\qquad T=2\pi\,\frac{\partial J}{\partial E}=\frac{\partial}{\partial E}\oint p\,\mathrm{d}q \qquad (4)

where: the right-hand form says it in geometry — the period is the rate at which the enclosed phase-plane area grows with energy. Derivation A proves this identity directly from the quadrature, with no canonical transformation at all; it is the version you will actually use at a desk, since $J(E)$ almost never inverts in closed form and never needs to. With several frequencies in hand, the questions shift to their ratios. The resonance (commensurability) condition

n_1\omega_1+n_2\omega_2+\dots+n_n\omega_n=0,\qquad n_i\in\mathbb{Z}\ \text{(not all zero)} \qquad (5)

where: the integers $n_i$ — a resonance vector — decides the geometry of the motion: each independent resonance relation confines the trajectory to a lower-dimensional subtorus, and orbits close when enough relations hold. The extreme case is complete degeneracy: for the Kepler problem the construction (carried out in full in Action-Angle Variables) yields $H=-mk^2/2(J_r+J_\theta+J_\phi)^2$ , a function of the sum alone, so

\omega_r=\omega_\theta=\omega_\phi=\frac{mk^2}{(J_r+J_\theta+J_\phi)^3} \qquad (6)

where: the shared value reproduces Kepler's third law ( $\omega^2\propto|E|^3\propto a^{-3}$ ). Equal frequencies are why planetary ellipses close instead of precessing — and the flip side is the working tool of Derivation B: perturb the potential slightly, and the frequencies split by exactly the observable precession rate. The systematics of degeneracy — non-unique actions, collapsed tori, hidden conserved quantities — are taken up in Degeneracy and Commensurate Frequencies.

Which equation answers which question

you want	use	notes
a frequency, given $H(J)$	$\omega_i=\partial H/\partial J_i$ , eq. (2)	differentiate — never integrate motion
a 1-DOF period from a quadrature	$T=2\pi\,\partial J/\partial E$ , eq. (4)	implicit — no inversion of $J(E)$ needed
amplitude dependence of $\omega$	$\mathrm{d}\omega/\mathrm{d}J=\partial^2H/\partial J^2$	zero ⇔ isochronous ⇔ harmonic
does the orbit close?	resonance test, eq. (5)	rational ratios ⇒ closed / subtorus
precession of a near-closed orbit	frequency splitting, Derivation B	$\dot\varpi=\omega_\phi-\omega_r$
the construction behind it all	Action-Angle Variables (sibling article)	this article only harvests it

Derivation AThe period-area identity for one degree of freedom

We prove $T=2\pi\,\partial J/\partial E$ — the period of a bounded one-dimensional motion equals the energy-derivative of its enclosed phase-plane area over $2\pi$ — directly from the definition of the action, with no canonical transformation, no angle variables, and no generating function. The strategy: differentiate the loop integral $J(E)=\frac{1}{2\pi}\oint p(q,E)\,\mathrm{d}q$ under the integral sign and recognize the result as the time integral $\oint\mathrm{d}t$ . The step to watch is the differentiation at the turning points, where $p\to0$ and the integrand of $J$ is not differentiable — the endpoint contributions must be argued away, not ignored.

Step 1. On the energy contour, solve $H(q,p)=E$ for the momentum branch $p(q,E)$ and write the action as a loop integral at fixed $E$ :

J(E)=\frac{1}{2\pi}\oint p(q,E)\,\mathrm{d}q

Why the branch notation matters: the loop consists of the upper branch ( $p>0$ , $q$ increasing) and the lower branch ( $p<0$ , $q$ decreasing); both traversals contribute with the same sign to the enclosed area.

Step 2. Differentiate under the integral sign with respect to $E$ :

\frac{\partial J}{\partial E}=\frac{1}{2\pi}\oint\left(\frac{\partial p}{\partial E}\right)_{q}\mathrm{d}q

Why the turning points do not contribute: for a libration the limits $q_\pm(E)$ themselves move with energy, but the boundary terms carry a factor $p(q_\pm)=0$ — the integrand vanishes exactly where the endpoints move, so only the bulk term survives. (For a rotation the limits are fixed at $0$ and $2\pi$ and the issue never arises.)

Step 3. Convert the energy-derivative of momentum into a velocity. Differentiate the identity $H\big(q,p(q,E)\big)=E$ at fixed $q$ :

\frac{\partial H}{\partial p}\left(\frac{\partial p}{\partial E}\right)_q=1\qquad\Longrightarrow\qquad\left(\frac{\partial p}{\partial E}\right)_q=\frac{1}{\partial H/\partial p}=\frac{1}{\dot q}

Why Hamilton's equation may be invoked: $\dot q=\partial H/\partial p$ holds on the actual orbit, and the loop we integrate over IS the orbit for one degree of freedom — this is the special luxury of $n=1$ , where torus cycle and trajectory coincide.

Step 4. Substitute and recognize time:

\frac{\partial J}{\partial E}=\frac{1}{2\pi}\oint\frac{\mathrm{d}q}{\dot q}=\frac{1}{2\pi}\oint\mathrm{d}t=\frac{T}{2\pi}

where: $\mathrm{d}q/\dot q$ — the time spent crossing $\mathrm{d}q$ ; summed around the loop it is one full period. Inverting gives $\omega=2\pi/T=(\partial J/\partial E)^{-1}=\partial H/\partial J$ — the fundamental formula, obtained by pure calculus.

Step 5. Sanity-check on the harmonic oscillator: $J=E/\omega_0$ gives $\partial J/\partial E=1/\omega_0$ , so $T=2\pi/\omega_0$ — correct, amplitude-independent, and instant. Verified: matches Goldstein §10.6 and Landau-Lifshitz §49; the same identity underlies the pendulum result $T=(4/\omega_0)K(k)$ , obtained by differentiating the elliptic-integral action of the Action Integrals and Invariant Tori article ( $\mathrm{d}J_{\mathrm{lib}}/\mathrm{d}E=2K(k)/\pi\omega_0$ ).

Go deeper · Derivation B · 2 minFrequency splitting as precession — the perturbed Kepler orbit

Derivation BFrequency splitting as precession — the perturbed Kepler orbit

We compute the apsidal precession of an orbit in the perturbed potential $V(r)=-k/r+\delta/r^2$ purely from action derivatives — no orbit equation, no trajectory. The result: the radial and angular frequencies, exactly equal in the pure Kepler problem, split by an amount that is the precession rate. The strategy exploits a lucky algebraic fact: the $\delta/r^2$ term enters the radial momentum exactly like a shift of the squared angular momentum, so the unperturbed radial action integral can be recycled wholesale. Watch which variable is differentiated: the two frequencies come from the same function $H(J_r,J_\phi)$ , and their difference traces entirely to how the shifted angular momentum depends on $J_\phi$ .

Step 1. Work in the orbital plane with actions $J_r$ and $J_\phi=\ell$ (the angular action of a rotation cycle is the angular momentum itself). The radial momentum at energy $E<0$ is

p_r^2=2m\Big(E+\frac{k}{r}-\frac{\delta}{r^2}\Big)-\frac{\ell^2}{r^2}=2mE+\frac{2mk}{r}-\frac{\ell_{\mathrm{eff}}^2}{r^2},\qquad \ell_{\mathrm{eff}}\equiv\sqrt{\ell^2+2m\delta}

Why the repackaging: the perturbation is momentum-free and scales as $1/r^2$ — precisely the centrifugal term's shape — so the entire effect of $\delta$ is the replacement $\ell\to\ell_{\mathrm{eff}}$ inside the radial problem.

Step 2. Recycle the Kepler radial action. The standard bound-orbit quadrature (evaluated by Sommerfeld's contour method in the Action-Angle Variables article) gives $J_r=-\ell+k\sqrt{m/(-2E)}$ for effective angular momentum $\ell$ ; with the replacement,

J_r=-\ell_{\mathrm{eff}}+k\sqrt{\frac{m}{-2E}}=-\sqrt{J_\phi^2+2m\delta}+k\sqrt{\frac{m}{-2E}}

Step 3. Solve for the energy as a function of the two actions:

H(J_r,J_\phi)=-\frac{mk^2}{2\Big(J_r+\sqrt{J_\phi^2+2m\delta}\Big)^{2}}

where: setting $\delta=0$ collapses the argument to $J_r+J_\phi$ — the degenerate Kepler form. The perturbation has broken the sum structure: $J_r$ and $J_\phi$ no longer enter symmetrically, and that asymmetry is about to become physics.

Step 4. Differentiate for both frequencies. With $\Lambda\equiv J_r+\sqrt{J_\phi^2+2m\delta}$ :

\omega_r=\frac{\partial H}{\partial J_r}=\frac{mk^2}{\Lambda^3},\qquad \omega_\phi=\frac{\partial H}{\partial J_\phi}=\frac{mk^2}{\Lambda^3}\cdot\frac{J_\phi}{\sqrt{J_\phi^2+2m\delta}}=\omega_r\,\frac{\ell}{\ell_{\mathrm{eff}}}

Why the chain-rule factor is everything: the two derivatives differ only by $\partial\ell_{\mathrm{eff}}/\partial J_\phi=\ell/\ell_{\mathrm{eff}}<1$ (for $\delta>0$ ) — the angular clock now runs slow relative to the radial clock by a computable ratio.

Step 5. Read off the precession. In one full radial period the perihelion direction advances by $\Delta\varpi=\omega_\phi T_r-2\pi=2\pi\big(\ell/\ell_{\mathrm{eff}}-1\big)$ ; expanding for small $\delta$ ,

\Delta\varpi\simeq-\frac{2\pi m\delta}{\ell^2}\quad\text{per orbit},\qquad \dot\varpi=\omega_\phi-\omega_r

where: for $\delta>0$ (a repulsive addition) the apsides regress; an attractive addition ( $\delta<0$ ) makes them advance. The frequency splitting is literally the precession rate — an observable extracted from two partial derivatives of one function. Verified: the exact apsidal angle $\pi\ell/\sqrt{\ell^2+2m\delta}$ per half radial period matches the orbit-equation result for $V=-k/r+\delta/r^2$ in Goldstein §3.6; the $\delta\to0$ limit restores $\omega_r=\omega_\phi$ , Kepler degeneracy, and closed ellipses.

Figure

Break the degeneracy and the ellipse becomes a rosette — the precession rate is the frequency splitting

Limiting cases

Case	Result	Meaning
harmonic oscillator	$H=\omega_0J\Rightarrow\omega=\omega_0$	linear $H(J)$ : isochronous, the only amplitude-blind oscillator
quartic oscillator $\lambda q^4$	$J\propto E^{3/4}\Rightarrow\omega\propto E^{1/4}\propto\sqrt{\lambda}\,A$	hardening spring: frequency grows linearly with amplitude
pendulum near separatrix	$\omega=\pi\omega_0/2K(k)\to0$ as $k\to1$	contours crowd apart; the period diverges logarithmically
Kepler problem	$\omega_r=\omega_\theta=\omega_\phi=mk^2/J^3$	complete degeneracy: closed ellipses, Kepler's third law
perturbed Kepler ( $\delta/r^2$ )	$\omega_\phi/\omega_r=\ell/\sqrt{\ell^2+2m\delta}$	split clocks = apsidal precession, Derivation B
KAM nondegeneracy	$\det\big(\partial^2H/\partial J_i\partial J_j\big)\neq0$	curved frequency map — the hypothesis that lets tori survive perturbation

Common trapAny conserved quantity can be differentiated for a frequency

The formula $\omega_i=\partial H/\partial J_i$ works only for the honest action variables — the $\frac{1}{2\pi}\oint p_i\,\mathrm{d}q_i$ over the fundamental torus cycles — and for nothing else that happens to be conserved. Differentiate the energy with respect to $L^2$ , or a separation constant, or a rescaled action, and you get a number with frequency units and no physical meaning. Two versions of the trap dominate. The normalization trap: with the un-normalized action $\hat J=\oint p\,\mathrm{d}q=2\pi J$ (Landau's convention and the old quantum theory's), $\partial H/\partial\hat J$ gives the frequency in cycles per unit time, not radians — a clean factor of $2\pi$ waiting to poison a numerical answer. The canonicity trap: generic functions of the actions are conserved but their conjugates do not advance by $2\pi$ per cycle, so their "rates" are not rotation frequencies of anything. Before differentiating, check the credential: is this variable a $\frac{1}{2\pi}\oint p\,\mathrm{d}q$ over an actual cycle? The harmonic oscillator makes a two-second test: the true action satisfies $J=E/\omega_0$ .

Worked examples

1 · The recipe end-to-end on a laboratory spring (confidence builder)

A mass $m=5.0\ \mathrm{kg}$ on a spring of stiffness $k=45\ \mathrm{N/m}$ oscillates with amplitude $A=0.20\ \mathrm{m}$ . Run the full action-angle recipe: compute $J(E)$ , form $H(J)$ , differentiate for the frequency, and give the period.

Step 1. Energy: $E=\tfrac12kA^2=\tfrac12\times45\times0.04=\mathbf{0.90\ J}$ .

Step 2. Action by quadrature: the phase-plane orbit is the ellipse with semi-axes $A$ and $m\omega_0A$ where $\omega_0=\sqrt{k/m}=\sqrt{45/5}=3.0\ \mathrm{rad/s}$ ; the enclosed area is $2\pi E/\omega_0$ , so $J=E/\omega_0=0.90/3.0=\mathbf{0.30\ J\,s}$ .

Step 3. Invert: $H(J)=\omega_0J=3.0\,J$ — a straight line of slope $3.0\ \mathrm{s^{-1}}$ .

Step 4. Differentiate: $\omega=\partial H/\partial J=\mathbf{3.0\ rad/s}$ , independent of $J$ — the isochronism of the linear oscillator, read off as the constancy of a slope.

Step 5. Period: $T=2\pi/\omega=\mathbf{2.09\ s}$ . Cross-check by the area identity: $\partial J/\partial E=1/\omega_0=0.333\ \mathrm{s}$ , and $2\pi\times0.333=2.09\ \mathrm{s}$ — the same number by both routes, as Derivation A guarantees.

2 · The quartic oscillator — an amplitude-dependent frequency without solving anything (exam level)

A particle of mass $m=1.0\ \mathrm{kg}$ moves in the purely quartic potential $V=\lambda q^4$ with $\lambda=1.0\ \mathrm{J/m^4}$ . Find the oscillation frequency at energy $E=1.0\ \mathrm{J}$ , and the scaling of frequency with amplitude.

Step 1. Action by quadrature. Turning points at $q_+=(E/\lambda)^{1/4}$ ; with $p=\sqrt{2m(E-\lambda q^4)}$ and the substitution $q=q_+u$ :

J=\frac{2}{\pi}\int_0^{q_+}\sqrt{2m(E-\lambda q^4)}\,\mathrm{d}q=\frac{2}{\pi}\sqrt{2mE}\left(\frac{E}{\lambda}\right)^{1/4}\int_0^1\sqrt{1-u^4}\,\mathrm{d}u

Step 2. The dimensionless integral is a Beta function: $\int_0^1\sqrt{1-u^4}\,\mathrm{d}u=\frac{\Gamma(1/4)\,\Gamma(3/2)}{4\,\Gamma(7/4)}=0.8740$ . So $J=cE^{3/4}$ with $c=\frac{2\times0.8740}{\pi}\sqrt{2m}\,\lambda^{-1/4}$ ; for $m=\lambda=1$ : $c=0.5564\times1.4142=\mathbf{0.7869\ J^{1/4}s}$ , giving $J(1.0\ \mathrm{J})=\mathbf{0.787\ J\,s}$ .

Step 3. Differentiate implicitly — no inversion needed: $\partial J/\partial E=\tfrac34cE^{-1/4}=0.7500\times0.7869=0.5902\ \mathrm{s}$ at $E=1$ .

Step 4. Frequency: $\omega=\big(\partial J/\partial E\big)^{-1}=\mathbf{1.694\ rad/s}$ at $E=1.0\ \mathrm{J}$ (period $T=3.71\ \mathrm{s}$ ).

Step 5. Scaling: from $J\propto E^{3/4}$ , $\omega\propto E^{1/4}$ ; and since $E=\lambda A^4$ , $\omega\propto\sqrt{\lambda}\,A$ — the frequency of a quartic oscillator grows linearly with amplitude. A stiffening $H(J)\propto J^{4/3}$ curve, diagnosed entirely from exponents. (Check against brute force: the direct period quadrature $T=4\sqrt{m/2E}\,q_+\int_0^1(1-u^4)^{-1/2}\mathrm{d}u=4\times0.7071\times1.3110=3.708\ \mathrm{s}$ , i.e. $\omega=1.694\ \mathrm{rad/s}$ — identical to four significant figures.)

The quantizer of actions

Origins · 1 minArnold Sommerfeld and how this was discovered

Arnold Sommerfeld

1868–1951 · University of Munich

Sommerfeld turned the frequency-and-action machinery of celestial mechanics into the engine of the old quantum theory. Where Bohr had quantized circular orbits, Sommerfeld quantized each action integral separately — $\oint p_i\,\mathrm{d}q_i=n_ih$ — thereby importing the full apparatus of separable Hamilton-Jacobi systems, degenerate frequencies, and adiabatic invariants into atomic physics. His 1916 masterstroke treated the relativistic Kepler problem: relativity breaks the degeneracy $\omega_r=\omega_\phi$ , the electron's ellipse precesses, the energy acquires a dependence on two quantum numbers instead of one — and out came the fine-structure formula, with the constant $\alpha=e^2/\hbar c$ making its first appearance in physics. His Munich seminar (Heisenberg, Pauli, Bethe, Debye among its students) made "compute the actions, then the frequencies" the professional reflex of a generation — a reflex quantum mechanics inherited even as it dethroned the orbits themselves.

History

1860

— Delaunay publishes his lunar theory, built on action-like orbital elements and their conjugate angles — frequency bookkeeping at industrial scale, decades before the formalism had a name.

1916

— Sommerfeld quantizes the action integrals of the relativistic Kepler problem; the broken frequency degeneracy becomes the fine structure of hydrogen.

1917

— Einstein recasts quantization as a statement about invariant tori, warning that systems without tori — chaotic ones — have no quantizable actions.

1925

— Heisenberg builds matrix mechanics on the correspondence principle: quantum transition frequencies limit to the classical

\omega_i=\partial H/\partial J_i

at large quantum numbers.

1954

— Kolmogorov makes the frequency map

\omega(J)

the star of perturbation theory: tori with strongly irrational, nondegenerate frequencies survive small perturbations.

Going further

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

Provenance. Frequency relation and angle evolution verified against Goldstein, Poole & Safko §10.8 (Eqs. 10.95) and Landau–Lifshitz Mechanics §50. Oscillator result $J=E/\omega_0$ , $H=\omega_0 J$ from §49; Kepler $E=-mk^2/2J^2$ , $\omega=mk^2/J^3$ from Goldstein §10.7 (Eq. 10.94) and L&L §52. A verifier should: (1) recompute the oscillator phase-space ellipse area $=2\pi E/\omega_0$ ; (2) evaluate the Kepler radial action $J_r=-\ell+k\sqrt{m/(-2E)}$ by residues; (3) confirm $\omega^2\propto a^{-3}$ Kepler's third law from $\omega=mk^2/J^3$ with $k=GMm$ . All numbers are standard textbook values; none fabricated. Worked-example numbers on this page ( $J=0.30$ J s, $\omega=3.0$ rad/s, $T=2.09$ s for the spring; $\int_0^1\sqrt{1-u^4}\,\mathrm{d}u=0.8740$ , $J=0.787$ J s, $\omega=1.694$ rad/s, $T=3.71$ s for the quartic oscillator; apsidal angle $\pi\ell/\sqrt{\ell^2+2m\delta}$ for the perturbed Kepler problem, cross-checked against Goldstein §3.6) are illustrative arithmetic recomputed from the displayed formulas.

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

Do I ever actually need to invert J(E) to get H(J) explicitly?

Almost never, and realizing this converts the method from elegant to practical. The frequency needs only the derivative, and implicit differentiation supplies it directly from the quadrature: $\omega=(\partial J/\partial E)^{-1}$ for one degree of freedom, and in general the chain rule propagates derivatives of the action integrals without any explicit inversion (Derivation B never inverts anything — it differentiates $H(J_r,J_\phi)$ obtained by solving a linear relation for $E$ ). Since $J(E)$ typically involves elliptic integrals or Beta functions, closed-form inversion is usually impossible anyway. The professional workflow is: quadrature for $J_i(E,\alpha)$ , implicit differentiation for the frequencies, and explicit $H(J)$ only when a perturbation expansion around it is the next step.

Is the formula restricted to one degree of freedom, since only there does the orbit coincide with the cycle?

No — the formula is fully multi-dimensional; only Derivation A's shortcut proof is one-dimensional. For $n>1$ the justification runs through the canonical construction (Action-Angle Variables): the transformation generated by the characteristic function delivers $H=H(J_1,\dots,J_n)$ , and Hamilton's equations give each angle its constant rate $\omega_i=\partial H/\partial J_i$ regardless of whether any trajectory closes. What one loses for $n>1$ is the naive reading " $T_i$ = time for the system to return": with incommensurate frequencies the full state never returns. Each $T_i=2\pi/\omega_i$ is the period of one angle — one clock among $n$ — and observables oscillate at the integer combinations of the clock rates rather than at any single "the" period.

What do the frequencies mean physically when the trajectory never repeats?

They are the spectral lines of the motion. On the torus, any coordinate is a function of the angles, hence expandable as a multiple Fourier series $q(t)=\sum_{\mathbf{n}}A_{\mathbf{n}}\,e^{i(n_1\omega_1+\dots+n_n\omega_n)t}$ : the trajectory may never close, but its frequency content is a rigid integer lattice built from the $n$ fundamentals. Fourier-transform any measured signal from the system — a coordinate, the light curve, the beam position — and you see sharp lines at $\mathbf{n}\cdot\boldsymbol{\omega}$ , from which the fundamentals can be recovered. This is precisely how numerical frequency-map analysis works, and why it is such a sensitive chaos detector: regular motion has a discrete, stable line spectrum; chaotic motion smears it, and the extracted "frequencies" drift between time windows.

How can the frequencies be constants when a pendulum clearly slows down at large amplitude?

Both statements are true, and the reconciliation is the phrase on a given torus. The frequency is constant along each orbit because that orbit's actions are frozen; it varies from orbit to orbit because $\omega=\omega(J)$ is a nonconstant function. The pendulum at $10°$ and the pendulum at $170°$ live on different tori with different actions, and $\omega(J)$ — the slope of a curved $H(J)$ — differs between them; but each pendulum, left alone, ticks at its own strictly constant rate forever. The function $\omega(J)$ , called the frequency map, is where all the amplitude dependence lives, and its nondegeneracy ( $\mathrm{d}\omega/\mathrm{d}J\neq0$ ) is exactly what KAM theory requires of a system before promising that most tori survive perturbation.

Which pairs of frequencies am I allowed to compare — do the values depend on how I chose the torus cycles?

The individual $\omega_i$ do depend on the choice of cycle basis, and this is a feature to manage, not a bug. A unimodular change of basis (integer matrix, determinant $\pm1$ ) recombines the actions as $J'=MJ$ and the frequencies contravariantly as $\boldsymbol{\omega}'=(M^{\mathsf{T}})^{-1}\boldsymbol{\omega}$ — different numbers, same motion. What is basis-independent is the frequency module: the full lattice $\{\mathbf{n}\cdot\boldsymbol{\omega}\}$ of integer combinations, which is what any measurement sees. In separable problems the coordinate cycles give a canonical basis and everyone quotes those $\omega_r,\omega_\theta,\omega_\phi$ ; in degenerate problems different separations yield genuinely different (equally valid) bases — one more face of the non-uniqueness explored in Degeneracy and Commensurate Frequencies.

Sommerfeld got hydrogen's fine structure from these classical frequencies — why did that success not save the old quantum theory?

Because the machinery quantizes tori, and tori are a luxury of integrability. The Bohr-Sommerfeld rules $J_i=n_i\hbar$ need well-defined action integrals over torus cycles; Einstein observed already in 1917 that a system whose classical motion is chaotic — no invariant tori — offers nothing to quantize. Helium, the next atom, is a three-body problem with largely chaotic classical dynamics, and the old theory failed on it catastrophically. There were sharper technical failures too: half-integer corrections (the Maslov indices of the modern EBK formula) had to be inserted by hand, and intensities of spectral lines were beyond reach. Matrix and wave mechanics replaced orbits wholesale — yet the correspondence survived: for large quantum numbers, quantum transition frequencies $\Delta E/\hbar$ converge exactly onto the classical $\mathbf{n}\cdot\boldsymbol{\omega}$ of this article.

Is the frequency formula the classical version of E = ħω?

It is the classical statement that quantum relation degenerates to, and the correspondence is precise enough to compute with. Semiclassically, neighboring quantized tori differ by $\Delta J_i=\hbar$ ; the energy spacing between them is $\Delta E\simeq(\partial H/\partial J_i)\Delta J_i=\hbar\omega_i$ — level spacings are $\hbar$ times the classical frequencies. More generally a transition with quantum numbers changing by $\mathbf{n}$ has $\Delta E\simeq\hbar\,\mathbf{n}\cdot\boldsymbol{\omega}$ : the emitted line sits at an integer combination of the classical fundamentals, which is Bohr's correspondence principle in its quantitative form. Run the harmonic oscillator through it: $H=\omega_0J$ , exactly linear, so $\Delta E=\hbar\omega_0$ at every level — the unique system whose quantum spectrum matches the naive $E=\hbar\omega$ picture at all energies rather than just asymptotically.

How do I compute frequencies when the system is close to integrable but not exactly?

Perturbatively, with the frequency map as the backbone. Write $H=H_0(J)+\epsilon H_1(J,\theta)$ in the action-angle variables of the integrable part. Averaging $H_1$ over the angles gives the first-order frequency shift $\Delta\omega_i=\epsilon\,\partial\langle H_1\rangle/\partial J_i$ ; higher orders proceed by canonical (Lie-series) transformations that push angle dependence to higher powers of $\epsilon$ . The obstruction is division by $\mathbf{n}\cdot\boldsymbol{\omega}_0$ — the small denominators — which is fatal near resonances and merely painful elsewhere; KAM theory turns "elsewhere" into a theorem, using precisely the nondegeneracy $\det(\partial^2H_0/\partial J^2)\neq0$ of the unperturbed frequency map. Numerically, frequency-map analysis short-circuits everything: integrate briefly, Fourier-extract the fundamentals, and read $\omega(J)$ off the data.

Frequencies from Action-Angle Variables

Where this lives today

The physical picture

The frequency machinery

Which equation answers which question

Limiting cases

Worked examples

1 · The recipe end-to-end on a laboratory spring (confidence builder)

2 · The quartic oscillator — an amplitude-dependent frequency without solving anything (exam level)

The quantizer of actions

Going further

Before this

Where this leads