Action-Angle Variables and Integrable Systems
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You must be comfortable with Hamilton's equations of motion, the concept of a canonical transformation, and the structure of the Hamilton-Jacobi equation as a method for finding constants of motion.
Why this matters
Action-angle variables transform complex, coupled oscillations into independent, uniform circular motions in phase space.
While Newtonian mechanics excels at predicting trajectories, it often struggles to reveal the underlying stability and periodicity of a system. By moving to action-angle variables, we decouple the "how much" energy is in a mode (the action) from the "where" in the cycle it currently is (the angle). This separation is not merely a mathematical convenience; it is the bridge to quantum mechanics, where actions are quantized, and to modern chaos theory, where the breakdown of these variables signals the onset of non-integrability.
The Geometric Intuition
Imagine a particle moving in a potential well. In standard position-momentum coordinates, the trajectory is a wiggly line that changes shape as energy changes. However, if you look at the area enclosed by this trajectory in phase space, you will notice something remarkable: for a periodic system, this area is fixed by the total energy (for the harmonic oscillator it is exactly proportional to it). This area is the "action." The "angle" is simply a clock that tells you where the particle is on its loop, progressing at a constant rate. By using these two variables, the complicated dynamics of a swinging pendulum or an orbiting planet become as simple as a point moving at constant speed around a circle. This simplification allows us to solve problems that are intractable with direct force analysis.
Defining the Action Variable
For a one-degree-of-freedom system with a periodic motion, the action variable \( J \) is defined as the area enclosed by the trajectory in the \( (q, p) \) phase plane. Mathematically, it is the integral of momentum over one complete cycle of the coordinate:
Since the motion is periodic, \( q \) returns to its starting value after a period \( T \). Because \( J \) is a constant of motion (it depends only on the energy \( E \)), it serves as a new momentum that does not change with time. This is a powerful result: we have found a canonical variable that is constant, which is the hallmark of a solved system. For a harmonic oscillator, this gives \( J = E/\omega \), so the action is simply a scaled version of the energy. (The factor \( 1/2\pi \) makes the conjugate angle advance by \( 2\pi \) per cycle; some books omit it and use an angle that advances by 1.)
The Angle Variable and Frequency
The conjugate variable to the action \( J \) is the angle \( \theta \). By the properties of canonical transformations, the Hamiltonian \( K \) expressed in terms of \( (J, \theta) \) depends only on \( J \), not on \( \theta \). Hamilton's equation for the angle is therefore:
Here, \( \omega(J) \) is the frequency of the motion. Since \( J \) is constant, \( \omega \) is constant, and the solution is simply \( \theta(t) = \omega t + \theta_0 \). The angle variable thus evolves linearly with time, representing uniform rotation in the new phase space. This is the essence of integrability: the complex nonlinear oscillation in \( q \) is mapped to a simple linear flow in \( \theta \).
Integrability and Multiple Degrees of Freedom
A system with \( n \) degrees of freedom is called integrable if it possesses \( n \) independent constants of motion in involution. In such systems, we can define \( n \) action variables \( J_1, \dots, J_n \) and \( n \) angle variables \( \theta_1, \dots, \theta_n \). The Hamiltonian becomes \( K(J_1, \dots, J_n) \), and the equations of motion decouple completely:
The motion is quasi-periodic, meaning it is a superposition of \( n \) independent periodic motions. If the frequencies \( \omega_i \) are commensurate (ratios of integers), the motion is strictly periodic. If they are incommensurate, the trajectory densely fills a torus in phase space. This geometric picture of motion on invariant tori is central to understanding stability in celestial mechanics and molecular dynamics.
The derivations
The Hamilton-Jacobi Equation
For a system with Hamiltonian H(q, p, t), there exists a canonical transformation to new coordinates that are all constants of motion, generated by Hamilton's principal function S(q1,…,qn, α1,…,αn, t). Requiring the transformed Hamiltonian to vanish yields the first-order partial differential equation H(q, ∂S/∂q, t) + ∂S/∂t = 0, whose complete integral reduces the dynamics to the algebraic inversion of 2n constants.
Canonical Transformations and Generating Functions
A change of phase-space variables (q,p) → (Q,P) preserves the form of Hamilton's equations if and only if there exists a generating function F such that p q̇ − H = P Q̇ − K + dF/dt. Choosing F to depend on one old and one new variable yields exactly four types — F1(q,Q,t), F2(q,P,t), F3(p,Q,t), F4(p,P,t) — whose partial derivatives supply the transformation equations and the new Hamiltonian K = H + ∂F/∂t.
F3(p,Q): q=−∂pF3, P=−∂QF3 • F4(p,P): q=−∂pF4, Q=∂PF4
and in all four cases K = H + ∂F/∂t
A common misconception is that action-angle variables exist for all bounded systems. They only exist for integrable systems. In non-integrable systems (like the three-body problem), the "torus" structure breaks down, and the action variables are no longer constants of motion. The failure of this formalism is a diagnostic tool for chaos. C
Worked example
Problem. Consider a one-dimensional harmonic oscillator with mass \( m \) and spring constant \( k \). The energy is \( E = \frac{1}{2} m \dot{q}^2 + \frac{1}{2} k q^2 \). Calculate the action variable \( J \) and the frequency \( \omega \) in terms of \( E \).
- Express momentum \( p \) in terms of \( q \) and \( E \): \( p = \pm \sqrt{2m(E - \frac{1}{2} k q^2)} \).
- Action over one cycle: \( J = \frac{1}{2\pi}\oint p\,dq = \frac{2}{2\pi}\int_{-A}^{A}\sqrt{2m\left(E - \tfrac{1}{2}kq^2\right)}\,dq \) with turning points \( A = \sqrt{2E/k} \).
- Since \( 2m(E - \tfrac12 kq^2) = mk(A^2 - q^2) \) and \( \int_{-A}^{A}\sqrt{A^2 - q^2}\,dq = \pi A^2/2 \): \( J = \frac{1}{\pi}\sqrt{mk}\,\frac{\pi A^2}{2} = \sqrt{mk}\,\frac{E}{k} = \frac{E}{\omega_0} \), with \( \omega_0 = \sqrt{k/m} \).
- Invert: \( K(J) = E = \omega_0 J \). The angle frequency is \( \omega = \partial K/\partial J = \omega_0 \), independent of amplitude, as expected for a harmonic oscillator.
Units check. \( J \) has units of \( \mathrm{J \cdot s} \) (action), which matches \( E \cdot T \). The frequency \( \omega \) has units of \( \mathrm{rad/s} \), consistent with \( \dot{\theta} \).
Three quick tests
- Is the action variable \( J \) a constant of motion for a periodic system?
Answer
Yes, by definition, \( J \) is a constant of motion in integrable systems. - What is the time dependence of the angle variable \( \theta \)?
Answer
It evolves linearly: \( \theta(t) = \omega t + \theta_0 \). - Can action-angle variables be defined for the three-body problem?
Answer
No, it is generally non-integrable, so the torus structure breaks down.
Problem set for PU-201 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-136 · D-135 · Threads: force, energy, symmetry