Action-angle coordinates

For one bounded periodic degree of freedom, integrate p dq once around the orbit in the direction of motion. The enclosed canonical area is positive for the ordinary oscillator convention. Dividing by two pi defines J with action units. Some books call the undivided integral the action variable instead; their corresponding angle has period one. Both choices work, but combining one book's action with another book's frequency formula introduces a factor-of-two-pi error.

For a harmonic oscillator, the phase-space ellipse has position semiaxis A and momentum semiaxis mωA. Its area is pi times their product. Energy is one half of mω squared times A squared, so the normalized action equals energy divided by angular frequency. This calculation uses the area of the closed curve, not the area under a position-time trace, and it does not mean that position and momentum are measured in identical units.

Choose position proportional to sine of the angle and momentum proportional to cosine. This orientation makes the angle increase at positive ω, and a direct bracket or symplectic-form check confirms the canonical sign. Expressing the oscillator Hamiltonian in these variables gives H equal to ωJ. Hamilton's equations then make J constant and the angle advance uniformly. The apparent complexity of sinusoidal position has become uniform motion around a circle in the angle coordinate.

At zero action the oscillator sits at its equilibrium. Every angle would represent the same point, so the angle coordinate is undefined there. This is a coordinate singularity, not a singular force. Even away from equilibrium, angle is understood modulo two pi rather than as a globally single-valued ordinary real coordinate. Using several overlapping branches, or treating it as a circle-valued quantity, avoids mistaking a jump from two pi to zero for a physical discontinuity.

For an n-degree-of-freedom integrable system, regular common level sets of n independent commuting integrals can support invariant tori when the necessary compactness and connectedness conditions hold. Locally near such a torus, actions are constant and angles evolve with frequencies obtained by differentiating H with respect to the actions. Several frequencies can be rationally related, giving closed motion, or incommensurate, giving motion that does not close after a finite period.

These coordinates are powerful because their existence is conditional. A separatrix is not an ordinary periodic torus: the period can diverge as it is approached, and the action-angle chart fails there. A generic nonintegrable system does not possess global action-angle coordinates covering every orbit. In later perturbation theory, actions remain useful starting variables, but resonances and broken invariant tori require analysis rather than an assumption that all motion has secretly become a set of independent oscillators.