Symplectic Structure and KAM Theory

Tier: R (Research). Builds on Hamiltonian mechanics and Noether’s theorem. This is where classical mechanics stops being a solved problem and becomes a live field: the KAM theorem is one of the deepest results in mathematical physics, and the questions it leaves open are still being worked on.

Recall prompts

Attempt these from memory before reading.

State Hamilton’s equations. What is the dimension of phase space for $N$ particles in three dimensions?
State Liouville’s theorem.
What is an integrable system? Give two examples.
What does the three-body problem have to do with King Oscar’s prize?

The four representations of a Hamiltonian system

Physics education research is unambiguous that experts move fluently between four representations; words, pictures, equations, and phase portraits; and that novices who are taught to translate between them outperform novices taught in equations alone. This page deliberately uses all four for the same object.

Take the plane pendulum.

In words. A mass on a massless rod swings under gravity. For small amplitudes it oscillates with fixed period; for large amplitudes it still oscillates, but slower; above a critical energy it goes over the top and rotates.

As equations. With $\theta$ the angle and $p_\theta = m\ell^2\dot\theta$ ,

H(\theta, p_\theta) = \frac{p_\theta^2}{2m\ell^2} - m g \ell \cos\theta, \qquad \dot\theta = \frac{\partial H}{\partial p_\theta},\quad \dot p_\theta = -\frac{\partial H}{\partial \theta}.

As a picture. The pendulum’s configuration space is a circle; its phase space is a cylinder $S^1 \times \mathbb{R}$ .

 p_theta
   ^
   |      rotation (unbounded)
   |   ...........................
 2mgl|..........................  separatrix
   |     /\        /\      /\
   |    /  \      /  \    /  \      oscillation
 0 +---/----\----/----\--/----\---.---> theta
   |            \  /    \/
   |             \/
   |      rotation (unbounded)

As a phase portrait. Closed curves around the origin for $H < 2mg\ell$ (oscillation), open curves for $H > 2mg\ell$ (rotation), and the separatrix between them, where the period diverges.

The separatrix is where the naive perturbation theory that KAM theory replaces breaks down, because the period $\int d\theta / \dot\theta$ diverges logarithmically. Keep that in mind: it is the reason the problem is hard.

The symplectic form: the structure that makes it all work

In words. Phase space is not just a set of points. It carries a geometric object; the symplectic form; that tells you how to measure area in a particular way, and Hamilton’s equations are exactly the statement that the flow preserves that area.

As equations. On $\mathbb{R}^{2n}$ with coordinates $(q_1,\dots,q_n,p_1,\dots,p_n)$ ,

\omega = \sum_{i=1}^n dq_i \wedge dp_i.

Hamilton’s equations are the single vector equation

\iota_{X_H}\,\omega = dH,

where $X_H$ is the vector field generating the flow. This compact form is why “symplectic” is the right generalisation and not “conservative”.

As a picture. Take any small parallelogram in phase space and evolve it forward under the flow.

      t = 0                    t = 1                    t = 2
     ......                   ......                   ......
    .      .                 .        .               .          .
   .        .    flow  →   .          .     flow  → .            .
    ........                 ..........               ............
     area = A                 area = A                 area = A
                          (shape distorts)         (shape distorts more)

As a theorem.

Theorem (Liouville, F/U). The flow of a Hamiltonian vector field preserves $\omega$ , and hence preserves the $2n$ -dimensional volume $\omega^n / n!$ .

This is why phase portraits of Hamiltonian systems never show spirals: an orbit spiralling inward would shrink area. Dissipative systems spiral; Hamiltonian systems do not. That single fact rules out the attractors that make dissipative chaos qualitatively different.

Exercise. Show that the divergence of the Hamiltonian vector field $\nabla \cdot X_H = \sum_i \left(\partial^2 H/\partial q_i \partial p_i - \partial^2 H/\partial p_i \partial q_i\right)$ vanishes, and relate this to Liouville’s theorem.

Poisson brackets: the algebra of observables

As equations.

\{f, g\} = \sum_{i=1}^n \left( \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i} \right)

and then Hamilton’s equations read, for any observable $f$ ,

\dot f = \{f, H\}.

Why this matters for research. The bracket makes the observables into a Lie algebra, and $\{q_i, p_j\} = \delta_{ij}$ is exactly the canonical commutation relation of quantum mechanics with $i\hbar$ times the bracket. The direction of quantisation runs from this structure, not from the Schrödinger equation; which is why deformation quantisation is formulated as a deformation of the Poisson algebra.

Integrable systems and action-angle variables

In words. A system with $n$ degrees of freedom is integrable if there are $n$ independent conserved quantities in involution. Then the motion is confined to an $n$ -torus and is quasi-periodic: the system is solvable, in a strong sense.

As equations. If $I = (I_1, \dots, I_n)$ satisfy $\{I_i, I_j\} = 0$ and the Hessian $\det(\partial^2 H/\partial I_i \partial I_j) \neq 0$ , there are angle coordinates $\theta_i$ on the tori with

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

The solution is then explicit:

\theta_i(t) = \theta_i(0) + \omega_i(I)\, t.

As a picture. The phase space fibres into invariant tori. On each torus the motion winds around, and whether the orbit is periodic or dense on the torus depends on whether the frequency ratios $\omega_i/\omega_j$ are rational.

Worked example (A). The harmonic oscillator has $H = \frac{1}{2}(p^2 + \omega^2 q^2)$ . With $I = H/\omega$ and $\theta = \arctan(\omega q / p)$ , one finds $H = \omega I$ and $\dot\theta = \omega$ ; a single torus (a circle), wound once per period.

The theorem this makes believable. The Arnold–Liouville theorem states the converse: $n$ involutive integrals on a compact $2n$ -dimensional level set imply the level set is a union of $n$ -tori with linear flow. Integrability is rare; the three-body problem is not integrable; so the question becomes what happens to the tori when the system is nearly integrable.

KAM theory: what survives a small perturbation

In words. Take an integrable system and perturb it slightly. Naive perturbation theory predicts the tori are destroyed, and for most tori they are. But Kolmogorov, Arnold and Moser showed that tori whose frequencies are irrational enough survive; distorted, but still invariant. A Cantor set of positive measure survives.

As equations. Take

H(I, \theta) = H_0(I) + \varepsilon H_1(I, \theta),

with $H_0$ non-degenerate. KAM says: for $|\varepsilon|$ small there is a Cantor set $K$ of invariant tori with measure tending to the unperturbed value as $\varepsilon \to 0$ , on which the motion remains quasi-periodic with frequencies satisfying a Diophantine condition

|\omega \cdot k| \geq \gamma |k|^{-\tau} \quad \text{for all } k \in \mathbb{Z}^n \setminus \{0\}.

Why the Diophantine condition. Perturbation theory produces series whose terms contain denominators $\omega \cdot k$ . For rational or nearly rational frequencies these get arbitrarily small and the series diverges. Diophantine frequencies are exactly those for which small divisors stay controlled; and the set of frequencies failing the condition has small measure. Number theory enters mechanics through the back door.

As a picture.

      integrable                 perturbed (KAM)
    ..............            ..............
   .    o    o    .          .    o    o    .
  .   o    o    o   .       .   x    o    o   .     o = torus survives
 .   o    o    o    .      .  o   x    o    o  .    x = torus destroyed,
 .    o    o    o   .      .   o    x    o    .        chaotic zone near it
  .    o    o    o  .       .   o    o    x   .
    ..............            ..............
   all tori survive          a Cantor set survives;
                             between them: chaos

Why it was hard. The proof requires a rapidly convergent iteration (Newton method rather than plain perturbation series) applied on a Cantor family of tori, with the parameters adjusted at each step. Kolmogorov’s 1954 scheme was completed by Arnold (1961–63) and, for area-preserving maps, Moser (1962) - the “KAM” initials.

Where the research frontier is.

Arnold diffusion. KAM tori do not fill phase space. In systems with $n \geq 3$ degrees of freedom, tori do not separate the energy surface, so orbits can drift between them along resonant channels. Arnold conjectured this in 1964; proving diffusion along a prescribed path and estimating its rate is an active field (Mather’s theory of Aubry–Mather sets, weak KAM theory).
Validity of KAM in solar system dynamics. Whether the solar system is actually stable on Gyr timescales is a numerical and rigorous question; Laskar’s work shows chaotic diffusion of the planetary orbits.
KAM for PDE. Extending KAM to infinite-dimensional systems (nonlinear Schrödinger, water waves) has been a major theme since the 1990s.

Counterexamples

KAM is not stability. The surviving tori have positive measure but empty interior. An orbit starting between two surviving tori is confined but not periodic, and in $n \geq 3$ may still drift. Reading KAM as “the system is stable” is the standard overstatement.

Small divisors are not a technicality. Poincaré showed the three-body problem’s perturbation series diverge. Every attempted “solution” of the $n$ -body problem by formal series is wrong, and Poincaré’s discovery of this is usually taken as the birth of chaos theory.

Ergodicity is not implied. A positive measure of surviving tori means the system is not ergodic on the energy surface. Treating KAM systems as ergodic; common in statistical mechanics reasoning; is unjustified.

Recall prompts: attempt after reading

State Liouville’s theorem, and say why Hamiltonian flow cannot spiral.
Write down the Poisson bracket and connect it to the canonical commutation relations.
What is an integrable system, and what does Arnold–Liouville conclude?
State the Diophantine condition and explain why small divisors force it.
What does KAM actually assert, and what does it not assert?

Interleaving problems

(Hamiltonian mechanics.) Verify the pendulum’s separatrix has energy $2mg\ell$ and compute the period near it, showing the logarithmic divergence.
(Noether’s theorem.) Show that a symmetry of $H$ generated by $G$ implies $\{G, H\} = 0$ , and connect this to $G$ being an integral.
(Measure theory.) Show the set of Diophantine frequencies has full measure for any $\tau > n - 1$ , and that the set of rationally dependent frequency vectors has measure zero.
(Real analysis.) Show the irrational rotation of the circle is minimal and uniquely ergodic, and that the rational rotation is neither.
(Complex analysis.) Where does the holomorphic functional calculus fail for the perturbed system, and how does that connect to the divergence of the perturbation series?

Symplectic Structure and KAM Theory