action-angle variables
Action-angle variables are the ideal coordinates for any system that repeats -- anything that oscillates or orbits. They repackage a bounded, periodic motion so that one variable (the angle) simply winds around at a steady rate, marking your phase around the cycle, while its partner (the action) stays perfectly constant, labeling which cycle you are on. They convert 'complicated periodic motion' into 'a point going around a circle at constant speed'.
For an integrable system whose motion is bounded, one performs the canonical transformation to new momenta called actions, J_i = (1/2 pi) times the closed-loop integral of p_i dq_i taken over one period of the i-th motion, and their conjugate coordinates the angles theta_i. In these variables the Hamiltonian depends on the actions alone, H = H(J), so the actions are all conserved and each angle evolves linearly: theta_dot_i = partial H / partial J_i = omega_i(J), a constant, giving theta_i = omega_i t + const. The omega_i are the natural frequencies of the motion, read off directly by differentiating H(J). All the hard work is in the one-time construction; afterwards the dynamics is trivially linear.
Action-angle variables are the natural language for perturbation theory, for adiabatic invariance (the actions are precisely the quantities that stay nearly constant under slow parameter changes), and for the old quantum theory, where Bohr-Sommerfeld quantization simply set each action to an integer multiple of Planck's constant, J_i = n_i hbar. They are also the stage on which the KAM theorem describes what survives when an integrable system is slightly perturbed. Whenever you hear 'frequencies' and 'tori', action-angle variables are underneath.
For the harmonic oscillator H = p^2/(2m) + (1/2) m omega^2 q^2, the action is J = E / omega (the phase-space area enclosed by an orbit, divided by 2 pi), so H = omega J. Then theta_dot = partial H / partial J = omega -- the angle advances at exactly the oscillator frequency, and the constant action is the enclosed area, energy divided by frequency.
For the oscillator the action is J = E/omega and the Hamiltonian is simply H = omega J, so the angle winds at the natural frequency omega.
Action-angle variables exist only for integrable systems with bounded (librating or rotating) motion; the construction relies on the phase-space trajectories lying on tori. A non-integrable or chaotic system has no global action-angle description -- the tori break up, and this breakdown is exactly what the KAM theorem and chaos theory study.