Canonical Transformations and Action-Angle Variables

Idea: Find "ideal" coordinates

Hamilton’s equations are elegant, but solving them in arbitrary coordinates is difficult. The goal of canonical transformations is to find coordinates (Q, P) in which the Hamiltonian takes the simplest form. In the best case — $\tilde{H}(Q, P)$ depends neither on $Q$ nor on $P$, and then the equations are trivial: $\dot{Q} = \text{const}$, $\dot{P} = 0$.

An analogy from school mathematics: when making a change of variables in integrals, we choose a convenient parametrization. In mechanics, the “convenient parametrization” is a canonical transformation.

Definition of Canonical Transformations

The transformation $(q, p) \to (Q, P)$ is called canonical if it preserves the form of Hamilton’s equations, that is, there exists $\tilde{H}(Q, P, t)$ such that $\dot{Q}_i = \partial \tilde{H}/\partial P_i$ and $\dot{P}_i = -\partial \tilde{H}/\partial Q_i$.

An equivalent condition: the transformation preserves the Poisson brackets: ${Q_i, P_j}{q,p} = \delta{ij}$.

Generating functions define a canonical transformation via linkage equations. There are four types:

Type $F_1(q, Q)$: $p_i = \partial F_1/\partial q_i, \quad P_i = -\partial F_1/\partial Q_i, \quad \tilde{H} = H + \partial F_1/\partial t$

Type $F_2(q, P)$: $p_i = \partial F_2/\partial q_i, \quad Q_i = \partial F_2/\partial P_i, \quad \tilde{H} = H + \partial F_2/\partial t$

Analysis: if $F_2 = \sum_i q_i P_i$ — this is the identity transformation $(Q = q, P = p)$. If $F_2 = \sum_i f_i(q) P_i$ — a coordinate transformation $Q = f(q)$.

Hamilton–Jacobi Equation

The idea: to find $F_2(q, P, t) = S(q, P, t)$ such that the new Hamiltonian $\tilde{H} = 0$. Then $\dot{P} = 0$, $\dot{Q} = 0$ — all new coordinates and momenta are constant, and the problem is solved!

The condition $\tilde{H} = 0$ when substituting $p_i = \partial S/\partial q_i$ gives the Hamilton–Jacobi equation:

$ \frac{\partial S}{\partial t} + H\left(q, \frac{\partial S}{\partial q}, t\right) = 0 $

This is a nonlinear partial differential equation for the function $S(q, t)$. Once $S$ is found, the problem reduces to algebra.

The physical meaning of $S$: it is the classical action along the optimal trajectory — $\int L, dt$. The quantum-mechanical Schrödinger equation, when substituting $\psi = \exp(iS/\hbar)$ and $\hbar \to 0$, reduces precisely to the HJE. Classical mechanics is the “high-frequency” limit of quantum!

Action-Angle Variables

For periodic systems, there exist special coordinates $(J, w)$, where $J$ is “action” (an invariant), $w$ is “angle” (a uniformly changing parameter).

The action variable (for a one-dimensional periodic system):

$ J = \frac{1}{2\pi} \oint p, dq $

The integral is taken over one period of motion in phase space. Unit: $[J] = \text{J}\cdot\text{s}$ (same as $\hbar$).

Conjugate angle $w$: $w = \partial S/\partial J$, velocity $\dot{w} = \partial H/\partial J = \omega(J)$ — constant.

Equations of motion in $(J, w)$: $\dot{J} = -\partial H/\partial w = 0$ ($J$ is constant!) and $\dot{w} = \omega(J) = \text{const}$. Solution: $J = \text{const}$, $w(t) = \omega t + w_0$. The action-angle variables completely “decouple” the equations of motion!

Full Numerical Example: Oscillator

For the harmonic oscillator $H = p^2/(2m) + m\omega^2 q^2/2$:

Step 1. Phase trajectory at energy $E$: ellipse $q^2/(2E/m\omega^2) + p^2/(2mE) = 1$

Step 2. $J = (1/2\pi) \oint p, dq = (1/2\pi) \cdot$ (area of ellipse) $= (1/2\pi)\cdot\pi\cdot\sqrt{2E/m\omega^2}\cdot\sqrt{2mE} = E/\omega$

Step 3. $E = J\omega \rightarrow H = J\omega \rightarrow \dot{w} = \partial H/\partial J = \omega = \text{const} ;\checkmark$

Step 4. Bohr–Sommerfeld quantization: $J = n\hbar \rightarrow E_n = n\hbar\omega$

This gives the correct oscillator spectrum! (The exact quantum answer: $E_n = \hbar\omega(n + 1/2)$, the correction $1/2$ is a quantum effect, “zero-point oscillations.”)

Real Application: Bohr–Sommerfeld Quantization

Before the creation of quantum mechanics (1920s) Bohr and Sommerfeld used the rule $J = n\hbar$ for quantizing atomic orbits. For the hydrogen atom: $J = (1/2\pi)\oint p_r dr + L_\phi$. Result: $E_n = -13.6,\text{eV}/n^2$ — correct hydrogen spectrum, explaining the Balmer, Lyman, and Paschen series.

The method was also applied to molecular rotation, explaining infrared spectra. Although it was replaced by full quantum mechanics, action-angle variables remain powerful tools in quantum field theory and chaos theory.

Adiabatic Invariants

The action variable $J$ is an adiabatic invariant: under slow (adiabatic) variation of system parameters $J$ is preserved, even if the energy changes. The classical example is a pendulum with gradually shortening string. As long as the length $l$ changes slowly, $J = E/\omega = \text{const}$, but $\omega = \sqrt{g/l}$ increases $\rightarrow$ energy $E = J\omega$ also increases proportionally to $\omega$. This explains why shortening the string “boosts” the pendulum: the work done by hand when pulling up the string is transferred to the kinetic energy of oscillations. In quantum mechanics, the theorem of adiabatic invariance underlies Berry's “geometric phase” (1984): as the parameters of the Hamiltonian change slowly adiabatically, the quantum state accumulates a geometric phase independent of the rate of change — only of the path in parameter space. This phase is observed experimentally in interferometry, photonic crystals, and topological insulators, forming a bridge between classical analytical mechanics and modern topological condensed matter physics.

Action–angle variables are applied in engineering for designing vibration protection systems: adiabatic variation of an oscillatory system parameter preserves $J$, allowing precise prediction of nonlinear mechanical oscillator behavior under slowly varying external conditions. In quantum chaos, Bohr–Sommerfeld rule $J = n\hbar$ is generalized to systems with several degrees of freedom through KAM tori, which is directly related to the stability of orbits in particle accelerators and the Solar System.

Exercise: Pendulum with energy $E < mgl$ (no full turns). (a) Compute $J = (1/2\pi) \oint p, d\theta$ via the elliptic integral. (b) For small oscillations: $J \approx E/\omega_0$, where $\omega_0 = \sqrt{g/l}$. Confirm Bohr quantization: $E_n = n\hbar\omega_0$. (c) How does $\omega$ depend on $J$ at large amplitudes? (Answer: $\omega$ decreases — a nonlinear oscillator is slower at larger amplitudes.)