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 $W(\mathbf{q})$ (not Hamilton's principal function $S$). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian $K(\mathbf{w}, \mathbf{J})$ is merely the old Hamiltonian $H(\mathbf{q}, \mathbf{p})$ expressed in terms of the new canonical coordinates, which we denote as $\mathbf{w}$ (the action angles, which are the generalized coordinates) and their new generalized momenta $\mathbf{J}$. We will not need to solve here for the generating function $W$ itself; instead, we will use it merely as a vehicle for relating the new and old canonical coordinates.

Rather than defining the action angles $\mathbf{w}$ directly, we define instead their generalized momenta, which resemble the classical action for each original generalized coordinate

$J_{k} \equiv \oint p_k \, dq_k$

where the integration path is implicitly given by the constant energy function $E=E(q_k,p_k)$. Since the actual motion is not involved in this integration, these generalized momenta $J_k$ are constants of the motion, implying that the transformed Hamiltonian $K$ does not depend on the conjugate generalized coordinates $w_k$

$\frac{d}{dt} J_{k} = 0 = \frac{\partial K}{\partial w_k}$

where the $w_k$ are given by the typical equation for a type-2 canonical transformation

$w_k \equiv \frac{\partial W}{\partial J_k}$

Hence, the new Hamiltonian $K=K(\mathbf{J})$ depends only on the new generalized momenta $\mathbf{J}$.

The dynamics of the action angles is given by Hamilton's equations

$\frac{d}{dt} w_k = \frac{\partial K}{\partial J_k} \equiv \nu_k(\mathbf{J})$

The right-hand side is a constant of the motion (since all the $J$'s are). Hence, the solution is given by

$w_k = \nu_k(\mathbf{J}) t + \beta_k$

where $\beta_k$ is a constant of integration. In particular, if the original generalized coordinate undergoes an oscillation or rotation of period $T$, the corresponding action angle $w_k$ changes by $\Delta w_k = \nu_k (\mathbf{J}) T$.

These $\nu_k(\mathbf{J})$ are the frequencies of oscillation/rotation for the original generalized coordinates $q_k$. To show this, we integrate the net change in the action angle $w_k$ over exactly one complete variation (i.e., oscillation or rotation) of its generalized coordinates $q_k$

$\Delta w_k \equiv \oint \frac{\partial w_k}{\partial q_k} \, dq_k = \oint \frac{\partial^2 W}{\partial J_k \, \partial q_k} \, dq_k = \frac{d}{dJ_k} \oint \frac{\partial W}{\partial q_k} \, dq_k = \frac{d}{dJ_k} \oint p_k \, dq_k = \frac{dJ_k}{dJ_k} = 1$

Setting the two expressions for $\Delta w_{k}$ equal, we obtain the desired equation

$\nu_k(\mathbf{J}) = \frac{1}{T}$

The action angles $\mathbf{w}$ are an independent set of generalized coordinates. Thus, in the general case, each original generalized coordinate $q_{k}$ can be expressed as a Fourier series in all the action angles

$q_k = \sum_{s_1=-\infty}^\infty \sum_{s_2 = -\infty}^\infty \cdots \sum_{s_N = -\infty}^\infty A^k_{s_1, s_2, \ldots, s_N} e^{i2\pi s_1 w_1} e^{i2\pi s_2 w_2} \cdots e^{i2\pi s_N w_N}$

where $A^k_{s_1, s_2, \ldots, s_N}$ is the Fourier series coefficient. In most practical cases, however, an original generalized coordinate $q_k$ will be expressible as a Fourier series in only its own action angles $w_k$

$q_k = \sum_{s_k=-\infty}^\infty e^{i2\pi s_k w_k}$

Summary of basic protocol

The general procedure has three steps:

1. Calculate the new generalized momenta $J_{k}$
2. Express the original Hamiltonian entirely in terms of these variables.
3. Take the derivatives of the Hamiltonian with respect to these momenta to obtain the frequencies $\nu_k$

Degeneracy

In some cases, the frequencies of two different generalized coordinates are identical, i.e., $\nu_k = \nu_l$ for $k \neq l$. 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.

See also

• Tautological one-form
• Integrable system
• Superintegrable Hamiltonian system

References

• L. D. Landau and E. M. Lifshitz, (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).
• H. Goldstein, (1980) Classical Mechanics, 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9
• G. Sardanashvily, (2015) Handbook of Integrable Hamiltonian Systems, URSS. ISBN 978-5-396-00687-4