Degeneracy and Commensurate Frequencies

Why is a Kepler orbit a closed ellipse and not a rosette? Why do hydrogen's energy levels crowd into families labelled by a single quantum number when three would seem to be needed? And why do Io, Europa and Ganymede orbit Jupiter with periods locked in the exact proportions 1:2:4? All three questions have the same answer, and it is arithmetic: whether the fundamental frequencies of an integrable system satisfy integer relations. The Liouville-Arnold Theorem delivers motion as uniform winding on a torus, with frequencies $\omega_i=\partial H/\partial J_i$ . What it does not say is how that winding looks — and here number theory takes over from geometry. Incommensurate frequencies (no integer relation $\sum_i m_i\omega_i=0$ ) wind the orbit densely over the whole torus, never repeating. Each independent integer relation — each resonance — flattens the motion onto a lower-dimensional subtorus, and when $n-1$ relations hold, every orbit closes into a one-dimensional loop. This is degeneracy, and it is never innocent: every resonance relation manufactures an extra conserved quantity beyond the $n$ actions, the fingerprint of a hidden symmetry. For Kepler that hidden constant is the Laplace-Runge-Lenz vector holding the perihelion fixed; for the isotropic oscillator, the Fradkin tensor. Degeneracy also marks the fault lines of mechanics. The "small denominators" $\mathbf m\cdot\boldsymbol\omega$ that plague perturbation theory vanish exactly on resonant tori, which is why those tori shatter first under perturbation — the opening move of KAM theory and chaos. This article develops the closure criterion, computes the Kepler degeneracy explicitly, extracts the hidden integrals, and follows the idea to Mercury's precessing perihelion and hydrogen's degenerate spectrum.

Key takeaways

On an invariant torus the motion has $n$ frequencies $\omega_i=\partial H/\partial J_i$ ; the orbit is closed precisely when all frequency ratios are rational, and densely fills the torus when no integer relation exists (Weyl equidistribution).
Each independent resonance relation $\sum_i m_i\omega_i=0$ makes the angle combination $\mathbf m\cdot\boldsymbol\theta$ a constant of motion — an extra integral beyond the actions, the signature of hidden symmetry.
Complete degeneracy (all frequencies equal, $n-1$ relations) closes every bounded orbit and yields $2n-1$ independent integrals — maximal superintegrability. Bertrand's theorem: among central potentials only $-k/r$ and $\tfrac12 kr^2$ achieve this at all energies.
Kepler is the archetype: the Hamiltonian depends on the actions only through $J_r+J_\theta+J_\varphi$ , so $\omega_r=\omega_\theta=\omega_\varphi$ — the ellipse never precesses, and the conserved perihelion direction is the Laplace-Runge-Lenz vector, generating a hidden $SO(4)$ .
Degeneracy is generically energy-dependent: anharmonic systems close their orbits only on measure-zero sets of tori. Kepler and the oscillator are special because their degeneracy is global.
Resonant tori are where perturbation-series denominators $\mathbf m\cdot\boldsymbol\omega$ vanish — the fragile tori that KAM theory sacrifices first, and where chaos is born.
Key idea: Closed orbits and reduced frequency counts both follow from one condition: the action-angle frequencies $\omega_i = \partial H/\partial J_i$ satisfy integer resonance relations $\sum_i m_i \omega_i = 0$ , and each independent relation hides an extra constant of motion.

Where this lives today

Resonance is the working vocabulary of modern celestial mechanics. The Kirkwood gaps in the asteroid belt sit at the 3:1, 5:2, 7:3 and 2:1 commensurabilities with Jupiter — resonant tori swept clean exactly as KAM predicts — while Pluto survives by resonance, its 3:2 lock with Neptune scheduling every close approach for maximum distance. Exoplanet astronomy inherited the whole toolkit: the TRAPPIST-1 system threads seven planets through a chain of near-commensurabilities whose librating phases (the frozen angles of equation 3) are used to weigh the planets, and "resonant chain" architectures are now read as fossils of slow disk migration. On the engineering side, accelerator physicists spend careers parking machine tunes away from low-order resonance lines, for exactly the small-denominator reasons this article derives. The quantum echo is just as loud. Classical complete degeneracy becomes quantum accidental degeneracy: hydrogen's levels depend only on the principal quantum number $n$ — the $n^2$ -fold crowding of states that makes the periodic table — because the quantized system inherits Kepler's $SO(4)$ ; the 3D oscillator's $SU(3)$ does the same for nuclear shell-model magic numbers. Lift the classical degeneracy and the quantum levels fan apart: the Lamb shift and fine structure are, in this language, quantum perihelion precessions. Farther afield, the Berry-Tabor conjecture makes degeneracy a spectral diagnostic — integrable systems' level spacings are Poissonian because their spectra are sums over tori with generically incommensurate frequencies — and resonance overlap (Chirikov's criterion) remains the standard back-of-envelope for the onset of chaos in everything from tokamak field lines to cold-atom kicked rotors.

The physical picture

Put a point on a 2-torus with two clocks, one per angle, ticking at rates $\omega_1$ and $\omega_2$ . If $\omega_1/\omega_2=p/q$ in lowest terms, the point comes home exactly after $q$ turns of the first clock and $p$ of the second: a closed curve that wraps the torus $(q,p)$ times — a Lissajous knot. If the ratio is irrational, home never recurs; the trajectory threads the torus forever, eventually passing arbitrarily close to every point, uniformly. Nothing about the dynamics distinguishes the two cases — both are linear winding — but the geometry of the trace could not be more different: a one-dimensional loop versus a dense two-dimensional smear.

Figure

Two windings of the same torus: a rational frequency ratio closes the orbit while an irrational ratio fills the surface densely

Degeneracy converts this arithmetic into conserved quantities. When two clocks tick in lockstep, their phase difference is frozen — a new constant of motion that no accounting of the actions predicted. For the Kepler ellipse the two clocks are the radial oscillation (perihelion to aphelion and back) and the angular revolution, and their lockstep means the perihelion always lands in the same place. The frozen phase is the perihelion direction, and its vector avatar is the Laplace-Runge-Lenz vector. Detune the clocks by any perturbation — an oblate planet, another body's pull, general relativity — and the frozen phase thaws: the perihelion creeps, the ellipse smears into a rosette, and the extra integral is gone.

Figure

Degeneracy holds the Kepler ellipse closed; lifting it lets the perihelion drift and the orbit becomes a rosette

The same lockstep, one level up, governs entire solar systems: three of Jupiter's moons share frozen phase relations (the Laplace resonance), asteroid belts are carved hollow at Jupiter-commensurate periods (the Kirkwood gaps), and Pluto survives crossing Neptune's orbit only because a 3:2 resonance choreographs their meetings. Commensurability is not a curiosity of special potentials; it is celestial mechanics' organizing principle — and, through small denominators, its principal headache.

The arithmetic of winding

The stage is set by the Liouville-Arnold Theorem: canonical action-angle coordinates $(J,\theta)$ in which the Hamiltonian is a function of the actions alone and each angle advances linearly,

\theta_i(t)=\omega_i t+\theta_i(0),\qquad \omega_i=\frac{\partial H}{\partial J_i} \qquad (1)

where: $J_i$ — the actions labelling the torus; $\omega_i$ — the fundamental frequencies, constant on each torus but generally varying from torus to torus. How to compute the $\omega_i$ in practice is the business of Frequencies from Action-Angle Variables; here we take them as given and ask what their ratios imply. Everything hinges on whether the frequency vector $\boldsymbol\omega$ admits a resonance:

\mathbf m\cdot\boldsymbol\omega=\sum_{i=1}^{n} m_i\,\omega_i=0,\qquad \mathbf m\in\mathbb{Z}^n\setminus\{0\} \qquad (2)

where: $\mathbf m$ — an integer vector, the resonance vector; the number $k$ of linearly independent solutions (the multiplicity of degeneracy) can range from $0$ (nondegenerate) to $n-1$ (completely degenerate). The condition is brutally sensitive: $\omega_1/\omega_2=1.5$ is resonant ( $\mathbf m=(2,-3)$ ), while $1.5+10^{-100}$ is not — which already hints that resonance is a measure-zero, dense, knife-edge phenomenon. Each resonance pays out a conserved quantity immediately. Consider the integer combination of angles $\phi_{\mathbf m}=\mathbf m\cdot\boldsymbol\theta$ ; differentiating along the flow (1),

\dot\phi_{\mathbf m}=\mathbf m\cdot\boldsymbol\omega=0 \qquad (3)

where: $\phi_{\mathbf m}$ — a genuine $2\pi$ -periodic phase-space function (integer coefficients keep it single-valued on the torus). A frozen angle is a constant of motion functionally independent of the actions — an integral the counting of Liouville Integrable Systems did not promise. Degenerate systems are therefore superintegrable: they carry more than $n$ integrals, though the extras can never join the commuting family (the isotropic bound caps involutive sets at $n$ ).

Derivation AClosed orbits require commensurate frequencies

We prove the closure criterion: a trajectory on an $n$ -torus with linear flow is periodic if and only if all frequency ratios are rational, and otherwise its closure is a subtorus whose dimension drops by one for each independent resonance. The strategy is elementary — impose exact recurrence and eliminate the period — with Weyl's equidistribution theorem supplying the density statement in the incommensurate case. Watch how little machinery is needed: the entire dichotomy between periodic and quasi-periodic motion is integer arithmetic.

Step 1. Suppose the orbit is closed with period $T$ : every angle must return to its start modulo full turns, $\theta_i(T)-\theta_i(0)=\omega_i T=2\pi n_i$ for integers $n_i$ , one per angle.

Step 2. Eliminate $T$ between any two components: $\omega_i/\omega_j=n_i/n_j\in\mathbb{Q}$ . Closure forces every pairwise ratio rational — equivalently, $n-1$ independent resonance relations hold. Conversely, if all ratios are rational, write $\omega_i=r_i\,\Omega$ with $r_i=p_i/q_i$ rational and $\Omega$ a common unit; then $T=2\pi\,\mathrm{lcm}(q_1,\dots,q_n)/\Omega\cdot$ (common factor) makes every $\omega_iT$ an integer multiple of $2\pi$ . Closed if and only if fully commensurate.

Step 3. Now the opposite extreme: no resonance at all, $\mathbf m\cdot\boldsymbol\omega\neq0$ for every nonzero integer vector. Weyl's equidistribution theorem then states that the sequence of angle positions is uniformly distributed on $\mathbb{T}^n$ : for any nice region $B$ , the fraction of time the orbit spends in $B$ converges to $\mathrm{vol}(B)/(2\pi)^n$ . In particular the orbit enters every open set — its closure is the whole torus. Why Weyl works: test uniformity against the Fourier basis $e^{i\mathbf m\cdot\boldsymbol\theta}$ ; the time average of $e^{i(\mathbf m\cdot\boldsymbol\omega)t}$ vanishes for every $\mathbf m\neq0$ exactly when no denominator $\mathbf m\cdot\boldsymbol\omega$ is zero — the same expressions that reappear as small denominators in perturbation theory.

Step 4. Partial resonance interpolates. Let $k$ be the number of independent integer relations. Each relation freezes one combination $\mathbf m\cdot\boldsymbol\theta$ (equation 3), confining motion to an $(n-k)$ -dimensional subtorus determined by the frozen values; on that subtorus the surviving $n-k$ effective frequencies are nonresonant, and Weyl fills it densely. The orbit's closure is a torus of dimension $n-k$ : from the full torus ( $k=0$ ) down to a closed loop ( $k=n-1$ ). Verified: reproduces Arnold §51 (conditionally periodic motion) and Goldstein §10.7; the $n=2$ case gives the familiar statement that a Lissajous figure closes exactly for rational frequency ratios.

Kepler: complete degeneracy made explicit

The claims so far are structural. The Kepler problem makes them computational — and spectacular. Separating the Hamilton-Jacobi equation in spherical coordinates (the machinery of Separation of Variables in Hamilton-Jacobi Theory) yields three one-dimensional action integrals, evaluated over each coordinate's libration or rotation cycle as in Action Integrals over Invariant Tori. We use the convention $J_i=\frac{1}{2\pi}\oint p_i\,\mathrm{d}q_i$ throughout, matching the sibling articles.

Go deeper · Derivation B · 1 minKepler degeneracy from the three action integrals

Derivation BKepler degeneracy from the three action integrals

We compute the actions of the bound Kepler problem ( $V=-k/r$ , $E<0$ ) and show the Hamiltonian collapses to a function of their sum alone, forcing all three frequencies equal — complete degeneracy, with closed ellipses and Kepler's third law tumbling out at the end. The strategy: evaluate the azimuthal and polar integrals by elementary means, quote the standard residue evaluation for the radial one, and watch the angular-momentum dependence cancel in the sum. The cancellation is the miracle to watch — it is the $1/r$ potential's private property.

Step 1. Azimuthal action. With $p_\varphi=L_z$ conserved and $\varphi$ running $0\to2\pi$ : $J_\varphi=\frac{1}{2\pi}\oint p_\varphi\,\mathrm{d}\varphi=L_z$ .

Step 2. Polar action. With $p_\theta^2=L^2-L_z^2/\sin^2\theta$ over the libration of $\theta$ between its turning points, the standard integral gives $J_\theta=\frac{1}{2\pi}\oint p_\theta\,\mathrm{d}\theta=L-|L_z|$ , where $L=|\mathbf L|$ . Why the combination matters already: $J_\theta+J_\varphi=L$ for prograde orbits — the polar and azimuthal actions know only the total angular momentum, not its orientation. The first cancellation has happened.

Step 3. Radial action. With $p_r^2=2mE+2mk/r-L^2/r^2$ between the radial turning points $r_\mp$ (perihelion and aphelion), the contour evaluation (Goldstein §10.8, by residues at $r=0$ and $r=\infty$ ) gives $J_r=\frac{1}{2\pi}\oint p_r\,\mathrm{d}r=-L+k\sqrt{\dfrac{m}{2|E|}}$ .

Step 4. Sum the three: $J_r+J_\theta+J_\varphi=-L+k\sqrt{\tfrac{m}{2|E|}}+L-|L_z|+|L_z|=k\sqrt{\dfrac{m}{2|E|}}$ . Every trace of $L$ and $L_z$ has cancelled: the total action depends on the energy alone.

Step 5. Invert for the Hamiltonian and differentiate:

H=-\frac{mk^2}{2\,(J_r+J_\theta+J_\varphi)^2}\qquad\Longrightarrow\qquad \omega_r=\omega_\theta=\omega_\varphi=\frac{mk^2}{(J_r+J_\theta+J_\varphi)^3}

All three partial derivatives are equal because $H$ sees only the sum — two independent resonance relations ( $\omega_r-\omega_\theta=0$ , $\omega_\theta-\omega_\varphi=0$ ), the maximum possible for $n=3$ : complete degeneracy. Every bound orbit closes after one radial period.

Step 6. Kepler's third law as a corollary. With $J_{\mathrm{tot}}=k\sqrt{m/2|E|}$ and $|E|=k/2a$ , the period is $T=2\pi/\omega=2\pi J_{\mathrm{tot}}^3/(mk^2)=\pi k\sqrt{\dfrac{m}{2|E|^3}}=2\pi\sqrt{\dfrac{m\,a^3}{k}}$ — i.e. $T^2\propto a^3$ . Verified: the energy-action relation matches Goldstein §10.8 and the worked satellite numbers of Liouville Integrable Systems ( $a=7.00\times10^6$ m gives $J_{\mathrm{tot}}=5.28\times10^{13}$ J·s and $T\approx97$ min); the circular-orbit limit $J_r=0$ reproduces $E=-mk^2/2L^2$ .

The frozen phases predicted by equation (3) have famous names here. Equality of $\omega_r$ and $\omega_\theta$ freezes $\theta_r-\theta_\theta$ , whose geometric meaning is the perihelion direction: the vector integral realizing it is the Laplace-Runge-Lenz vector

\mathbf A=\mathbf p\times\mathbf L-mk\,\hat{\mathbf r},\qquad \frac{\mathrm{d}\mathbf A}{\mathrm{d}t}=0 \qquad (4)

where: $\mathbf A$ — conserved for the pure $1/r$ force only; it lies in the orbit plane, points at perihelion, and has magnitude $|\mathbf A|=mke$ with $e$ the eccentricity. Counting integrals: energy, three components of $\mathbf L$ , three of $\mathbf A$ , minus two functional relations ( $\mathbf A\cdot\mathbf L=0$ and $A^2=m^2k^2+2mEL^2$ ) — five independent integrals, $2n-1$ for $n=3$ : maximal superintegrability, the algebraic face of every-orbit-closes. The Poisson brackets of $\mathbf L$ with the rescaled $\tilde{\mathbf A}=\mathbf A/\sqrt{2m|E|}$ close into the Lie algebra $\mathfrak{so}(4)$ for bound motion — the hidden four-dimensional rotational symmetry that Pauli would later use to crack hydrogen's spectrum before wave mechanics existed. The isotropic oscillator plays the same game with different cards: $H=\omega(J_1+J_2+J_3)$ makes all frequencies equal identically, the conserved Fradkin tensor $A_{ij}=\tfrac{1}{2m}p_ip_j+\tfrac12 m\omega^2x_ix_j$ replaces $\mathbf A$ , and the symmetry algebra is $\mathfrak{su}(3)$ . Bertrand's theorem (1873) crowns the story: among all central potentials, exactly two — $V=-k/r$ and $V=\tfrac12kr^2$ — make every bounded orbit closed at every energy. The proof pits the apsidal angle (computable from the frequencies) against the requirement that it be a fixed rational multiple of $\pi$ across all orbits; only the exponents $-1$ and $+2$ survive. Global degeneracy is not a family of examples; it is two isolated jewels.

Limiting cases

Case	Result	Meaning
No resonance ( $k=0$ )	orbit dense on $\mathbb{T}^n$	Nondegenerate: quasi-periodic motion, exactly $n$ frequencies, no extra integrals; the generic torus.
Partial degeneracy ( $1\le k<n-1$ )	orbit dense on $\mathbb{T}^{n-k}$	$k$ frozen phases = $k$ extra integrals; superintegrable but orbits do not all close.
Complete degeneracy ( $k=n-1$ )	every bounded orbit closed	$2n-1$ integrals, maximal superintegrability, hidden symmetry algebra (Kepler $SO(4)$ , oscillator $SU(3)$ ).
Energy-dependent resonance	closure only at special $J$	Generic anharmonic systems: rational ratios on a dense measure-zero set of tori; orbits precess at almost every energy.
Perturbed resonance	$\mathbf m\cdot\boldsymbol\omega\approx0$	Small denominators: perturbation series blow up, resonant tori shatter into island chains — the KAM frontier.

Common trapBounded orbits eventually close on themselves

They almost never do. Boundedness confines an integrable orbit to a torus; closure additionally requires every frequency ratio to be rational, which for a generic system holds only on a measure-zero set of tori. The typical bounded orbit of a typical integrable system is a quasi-periodic space-filler — a rosette, not an ellipse. Kepler and the isotropic oscillator are the only central-force exceptions (Bertrand), which is precisely why the solar system's closed ellipses demanded explanation and why any observed precession (Mercury's 43 arcseconds per century) is a direct measurement of degeneracy-breaking physics. When you sketch a generic bound orbit as a neat closed loop, you are silently invoking one of the two most special potentials in mechanics.

Worked examples

Example 1 — the Laplace resonance of Io, Europa and Ganymede (confidence builder). Jupiter's three inner Galilean moons have observed mean motions (average angular frequencies) $n_1=203.489^\circ/\mathrm{day}$ (Io), $n_2=101.375^\circ/\mathrm{day}$ (Europa), $n_3=50.318^\circ/\mathrm{day}$ (Ganymede). Test the moons for a three-body resonance with vector $\mathbf m=(1,-3,2)$ .

Step 1. Form the resonance combination: $\mathbf m\cdot\mathbf n=n_1-3n_2+2n_3=203.489-3(101.375)+2(50.318)$ .

Step 2. Evaluate: $3\times101.375=304.125$ and $2\times50.318=100.636$ , so $\mathbf m\cdot\mathbf n=203.489-304.125+100.636=\mathbf{0.000^\circ/day}$ — zero to the full precision of the data. The three frequencies satisfy an exact integer relation even though no pair of them is exactly commensurate ( $n_1/n_2=2.0073$ , not $2$ ).

Step 3. Extract the frozen phase. By equation (3) the angle combination $\phi=\lambda_1-3\lambda_2+2\lambda_3$ (mean longitudes) is a constant of motion; dynamically it librates about $180^\circ$ — the configuration in which all three moons can never meet on the same side of Jupiter. One resonance relation, one conserved phase, exactly as the theory orders.

Step 4. Read off the periods to see the famous ratios: $T_i=360^\circ/n_i$ gives $T_1=1.769$ d, $T_2=3.551$ d, $T_3=7.155$ d — the 1:2:4 chain. The lock is maintained dynamically (tidal dissipation pumps energy while the resonance steers it), which is why it has survived for billions of years: an astronomical-scale demonstration that resonant phases are physical, protectable quantities.

Example 2 — Mercury: measuring broken degeneracy in arcseconds (exam level). General relativity perturbs the Sun's $1/r$ potential, detuning $\omega_\theta$ from $\omega_r$ by the per-orbit perihelion advance $\Delta\varphi=6\pi GM_\odot/[c^2a(1-e^2)]$ . For Mercury, $a=5.79\times10^{10}$ m, $e=0.2056$ , $GM_\odot=1.327\times10^{20}\ \mathrm{m^3/s^2}$ , orbital period $T=87.97$ d. Compute the century precession rate.

Step 1. Denominator: $c^2a(1-e^2)=(8.988\times10^{16})(5.79\times10^{10})(1-0.0423)=(8.988\times10^{16})(5.79\times10^{10})(0.9577)=4.98\times10^{27}\ \mathrm{m^3/s^2}$ .

Step 2. Per-orbit advance: $\Delta\varphi=\dfrac{6\pi(1.327\times10^{20})}{4.98\times10^{27}}=\dfrac{2.501\times10^{21}}{4.98\times10^{27}}=\mathbf{5.02\times10^{-7}\ rad/orbit}$ — the two clocks slip by half a microradian per lap.

Step 3. Orbits per century: $N=\dfrac{100\times365.25}{87.97}=415.2$ .

Step 4. Century drift: $\Delta\varphi_{100}=415.2\times5.02\times10^{-7}=2.08\times10^{-4}$ rad. Convert: $2.08\times10^{-4}\times206{,}265\ \mathrm{arcsec/rad}=\mathbf{43.0''\ per\ century}$ — the classic anomaly Le Verrier isolated in 1859 and Einstein explained in 1915. The measurement is, word for word, a measurement of how badly general relativity breaks the Kepler resonance $\omega_r=\omega_\theta$ : the conserved LRL vector of the $1/r$ problem becomes a slowly rotating hand on the solar clock, turning once every $2\pi/\Delta\varphi\approx1.25\times10^{7}$ orbits — three million years.

The history band

Origins · 1 minJürgen Moser and how this was discovered

Jürgen Moser

1928–1999 · escaped East Prussia on foot, then tamed the resonances that Poincaré feared — the M of KAM

Moser crossed the collapsing eastern front as a teenager and reached Göttingen with nothing; by 1962 he had supplied the hardest missing piece of the perturbation problem this article's resonances create. Kolmogorov's 1954 announcement — that strongly irrational tori survive perturbation — assumed analytic Hamiltonians; Moser's twist theorem proved survival with only finitely many derivatives, turning a miracle of analyticity into robust mathematics, and the acronym KAM acquired its M. His stability results govern exactly the landscape mapped here: resonant tori (this article's degenerate ones) are destroyed, Diophantine tori persist, and the boundary between them is drawn by the arithmetic of $\mathbf m\cdot\boldsymbol\omega$ . Moser later found the integrable geodesic flow on the ellipsoid hiding inside the same circle of ideas, and spent a career shuttling between celestial mechanics and pure analysis — Poincaré's heir on both counts.

History

1805

— Laplace explains the 1:2:4 lock of Io, Europa and Ganymede: the first analyzed three-body resonance

1859

— Le Verrier isolates Mercury's anomalous perihelion drift of 43 arcseconds per century

1873

— Bertrand proves only the 1/r and r-squared potentials close all bounded orbits

1915

— Einstein computes Mercury's 43 arcseconds from general relativity — degeneracy-breaking measured

1926

— Pauli solves hydrogen algebraically from the Runge-Lenz SO(4), before the Schrödinger equation

1954

— Kolmogorov announces persistence of nonresonant tori; Arnold and Moser complete the proofs by 1963

Going further

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §10.7–10.8 and §3.9 (LRL vector) · Landau & Lifshitz, Mechanics, 3rd ed., §49–52 · Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., §51 (conditionally periodic motion) · Taylor, Classical Mechanics, Ch. 8 · Murray & Dermott, Solar System Dynamics, Ch. 8 (resonances)

Provenance. Action integrals and the Kepler Hamiltonian $H=-mk^2/[2(J_r+J_\theta+J_\varphi)^2]$ (in the $\frac{1}{2\pi}\oint$ convention) should be recomputed against Goldstein §10.8; the closure criterion (rational ratios ⇔ closed orbits, Weyl density otherwise) against Arnold §51 and Landau-Lifshitz §52. The LRL vector $\mathbf A=\mathbf p\times\mathbf L-mk\hat{\mathbf r}$ and the $SO(4)$ algebra are standard (Goldstein §3.9). Numeric checks computed for this article and hand-verifiable: Laplace resonance $n_1-3n_2+2n_3=203.489-304.125+100.636=0.000^\circ$ /day from published mean motions; Mercury per-orbit advance $6\pi GM_\odot/[c^2a(1-e^2)]=5.02\times10^{-7}$ rad and $415.2$ orbits/century giving $43.0''$ /century.

Builds on: Action-Angle Variables · Frequencies from Action-Angle Variables · Liouville-Arnold Theorem · Separation of Variables in Hamilton-Jacobi Theory

Leads to: Action Integrals over Invariant Tori · Liouville Integrable Systems · Staeckel Conditions for Separability

Rational numbers are dense — is not every torus arbitrarily close to a resonant one, making the distinction useless?

Dense, yes; typical, no — and the distinction is exactly what KAM theory quantifies. The resonant tori form a dense set of measure zero: pick action values at random and with probability one the frequencies are incommensurate. What matters dynamically is not resonance versus non-resonance but how close to resonance, measured by how small $|\mathbf m\cdot\boldsymbol\omega|$ can get relative to $|\mathbf m|$ . Diophantine frequency vectors — those with $|\mathbf m\cdot\boldsymbol\omega|\ge\gamma/|\mathbf m|^\tau$ — stay quantifiably far from all resonances and fill most of measure; their tori survive perturbation. The dense resonant skeleton, plus a thin layer around it, is destroyed. Density of rationals makes the surviving set a fractal-like Cantor family — but a fat one.

Is degeneracy the same thing as superintegrability?

They are two faces of one phenomenon, in different languages. Degeneracy is dynamical: integer relations among the torus frequencies. Superintegrability is algebraic: more than $n$ functionally independent integrals of motion. Equation (3) is the bridge — each independent resonance relation freezes one angle combination, minting one extra integral, so $k$ relations give $n+k$ integrals. Maximal degeneracy ( $k=n-1$ ) means maximal superintegrability ( $2n-1$ integrals) and all orbits closed. One caution: the extra integrals cannot commute with everything (the involutive cap is $n$ ), so a superintegrable system has a non-abelian symmetry algebra — $\mathfrak{so}(4)$ for Kepler, $\mathfrak{su}(3)$ for the oscillator — which is why its quantum version shows degenerate multiplets rather than extra quantum numbers.

Why is hydrogen's degeneracy called accidental when it clearly has a cause?

The name is a historical fossil worth decoding. Rotational symmetry alone explains why energies cannot depend on the magnetic quantum number $m$ — that degeneracy is "geometric" and expected. But rotations say nothing about $\ell$ : levels with different orbital shapes should split, and in every non-Coulomb central potential (alkali atoms, for instance) they do. Hydrogen's extra $\ell$ -degeneracy looked unexplained — accidental — until Pauli (1926) and Fock (1935) traced it to the quantized Runge-Lenz vector and the hidden $SO(4)$ . Classically the same cause makes $\omega_r=\omega_\theta$ and closes the ellipses. So "accidental" now means precisely: caused by a dynamical symmetry larger than the obvious geometric one, i.e. by the degeneracy structure this article develops. The accident was in our bookkeeping, not in nature.

If resonant tori are the first destroyed by perturbations, how has the Laplace resonance survived for billions of years?

Because "resonant tori shatter" is a statement about Hamiltonian, conservative perturbation theory, and the Galilean moons are not a conservative system: Jupiter's tides continuously drain and pump orbital energy. Dissipation converts the resonance from a fragile torus into an attractor — the moons were slowly migrated into the 1:2:4 lock by tidal evolution, and the same dissipation damps any libration of the frozen phase back toward its equilibrium at $180^\circ$ . What is destroyed in the conservative problem is the invariant torus as an exact barrier; what survives in the dissipative problem is a stable locked state guarded by an effective potential well in the resonant angle. The same physics parks exoplanet chains (TRAPPIST-1) and spun Mercury into its 3:2 spin-orbit resonance. KAM tells you where the cliffs are; dissipation decides whether systems fall off or nestle against them.

Does the Runge-Lenz vector survive for unbound Kepler orbits, and what happens to the symmetry algebra?

$\mathbf A$ is conserved for any orbit in a pure $1/r$ potential — the derivation never uses the sign of $E$ . For hyperbolic orbits it points along the symmetry axis of the branch and its magnitude still encodes the eccentricity ( $e>1$ ). What changes is the algebra: the natural rescaling of $\mathbf A$ involves $\sqrt{2m|E|}$ , and for $E>0$ the brackets of $\mathbf L$ with $\tilde{\mathbf A}=\mathbf A/\sqrt{2mE}$ close into $\mathfrak{so}(3,1)$ — the Lorentz algebra — instead of $\mathfrak{so}(4)$ . The quantum reflection of this is that hydrogen's continuum states organize into representations of the Lorentz group, while the bound spectrum realizes $SO(4)$ . One conserved vector, two symmetry algebras, split by the sign of the energy: a first sighting of non-compact dynamical symmetry.

What sets the width of a resonance when a perturbation is switched on?

Near a resonance $\mathbf m\cdot\boldsymbol\omega\approx0$ , transform to the slow angle $\psi=\mathbf m\cdot\boldsymbol\theta$ and average over the fast ones: the surviving dynamics is a pendulum, $\ddot\psi\approx-\varepsilon F\sin\psi$ , where $\varepsilon F$ is the strength of the resonant Fourier component of the perturbation. The libration zone of that pendulum — the island chain in a Poincaré section — has half-width in frequency $\Delta\omega\sim\sqrt{\varepsilon}$ , i.e. in action $\Delta J\sim\sqrt{\varepsilon F/(\partial\omega/\partial J)}$ . The square root is the important scaling: even tiny perturbations open finite resonance zones. Chirikov's criterion then estimates chaos onset as resonance overlap: when the $\sqrt{\varepsilon}$ -widths of neighbouring island chains touch, trajectories wander between them and the last KAM tori in the gap die. That back-of-envelope governs tokamak field-line stochasticity and the chaotic rotation of Hyperion alike.

Are there systems with degeneracy at some energies but not others, and what does the transition look like?

That is the generic situation — global degeneracy is the exception confined to Bertrand's two potentials. In a typical anharmonic system the frequency ratio $\omega_1(J)/\omega_2(J)$ varies continuously with the actions, sweeping through rational values on a dense set of tori: each rational crossing is a torus whose orbits close, embedded among tori whose orbits precess. Nothing dramatic happens in the integrable system itself — closure at isolated tori is invisible to smooth observables. The drama arrives with perturbation: precisely those rational-ratio tori sprout island chains of width $\sqrt{\varepsilon}$ , so a Poincaré section of the perturbed system is a map of the unperturbed degeneracies. Scanning energy in an experiment (a Rydberg atom in a field, a particle accelerator ramping tune) drags the system across resonance after resonance, each announcing itself by a burst of structure — the practical signature of energy-dependent commensurability.