Action-angle coordinates

In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates are chiefly used when the Hamilton–Jacobi equations are completely separable. (Hence, the Hamiltonian does not depend explicitly on time, i.e., the energy is conserved.) Action-angle variables define an invariant torus, so called because holding the action constant defines the surface of a torus, while the angle variables parameterize the coordinates on the torus.
The Bohr–Sommerfeld quantization conditions, used to develop quantum mechanics before the advent of wave mechanics, state that the action must be an integral multiple of Planck's constant; similarly, Einstein's insight into EBK quantization and the difficulty of quantizing non-integrable systems was expressed in terms of the invariant tori of action-angle coordinates.
Action-angle coordinates are also useful in perturbation theory of Hamiltonian mechanics, especially in determining adiabatic invariants. One of the earliest results from chaos theory, for the non-linear perturbations of dynamical systems with a small number of degrees of freedom is the KAM theorem, which states that the invariant tori are stable under small perturbations.
The use of action-angle variables was central to the solution of the Toda lattice, and to the definition of Lax pairs, or more generally, the idea of the isospectral evolution of a system.
Derivation.
Action angles result from a type-2 canonical transformation where the generating function is Hamilton's characteristic function formula_1 ("not" Hamilton's principal function formula_2). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian formula_3 is merely the old Hamiltonian formula_4 expressed in terms of the new canonical coordinates, which we denote as formula_5 (the action angles, which are the generalized coordinates) and their new generalized momenta formula_6. We will not need to solve here for the generating function formula_7 itself; instead, we will use it merely as a vehicle for relating the new and old canonical coordinates.
Rather than defining the action angles formula_5 directly, we define instead their generalized momenta, which resemble the classical action for each original generalized coordinate
where the integration path is implicitly given by the constant energy function formula_10. Since the actual motion is not involved in this integration, these generalized momenta formula_11 are constants of the motion, implying that the transformed Hamiltonian formula_12 does not depend on the conjugate generalized coordinates formula_13
where the formula_13 are given by the typical equation for a type-2 canonical transformation
Hence, the new Hamiltonian formula_17 depends only on the new generalized momenta formula_6.
The dynamics of the action angles is given by Hamilton's equations
The right-hand side is a constant of the motion (since all the formula_20's are). Hence, the solution is given by
where formula_22 is a constant of integration. In particular, if the original generalized coordinate undergoes an oscillation or rotation of period formula_23, the corresponding action angle formula_13 changes by formula_25.
These formula_26 are the frequencies of oscillation/rotation for the original generalized coordinates formula_27. To show this, we integrate the net change in the action angle formula_13 over exactly one complete variation (i.e., oscillation or rotation) of its generalized coordinates formula_27
Setting the two expressions for formula_31 equal, we obtain the desired equation
The action angles formula_5 are an independent set of generalized coordinates. Thus, in the general case, each original generalized coordinate formula_34 can be expressed as a Fourier series in "all" the action angles
where formula_36 is the Fourier series coefficient. In most practical cases, however, an original generalized coordinate formula_27 will be expressible as a Fourier series in only its own action angles formula_13
Summary of basic protocol.
The general procedure has three steps:
Degeneracy.
In some cases, the frequencies of two different generalized coordinates are identical, i.e., formula_42 for formula_43. In such cases, the motion is called degenerate.
Degenerate motion signals that there are additional general conserved quantities; for example, the frequencies of the Kepler problem are degenerate, corresponding to the conservation of the Laplace–Runge–Lenz vector.
Degenerate motion also signals that the Hamilton–Jacobi equations are completely separable in more than one coordinate system; for example, the Kepler problem is completely separable in both spherical coordinates and parabolic coordinates.