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 provide 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 is over all possible values of $q\_\{k\}$, given the energy $E$. 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

:$frac\{d\}\{dt\}\; w\_\{k\}\; =\; frac\{partial\; K\}\{partial\; J\_\{k\; equiv\; u\_\{k\}(mathbf\{J\})$

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

:

where 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 $Delta\; w\_\{k\}\; =\; u\_\{k\}(mathbf\{J\})\; T$.

These $u\_\{k\}(mathbf\{J\})$ 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

:$u\_\{k\}(mathbf\{J\})\; =\; frac\{1\}\{T\}$

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

:$q\_\{k\}\; =\; sum\_\{s\_\{1\}=-infty\}^\{infty\}\; sum\_\{s\_\{2\}=-infty\}^\{infty\}\; ldots\; sum\_\{s\_\{N\}=-infty\}^\{infty\}\; A^\{k\}\_\{s\_\{1\},\; s\_\{2\},\; ldots,\; s\_\{N\; e^\{i2pi\; s\_\{1\}\; w\_\{1\; e^\{i2pi\; s\_\{2\}\; w\_\{2\; ldots\; e^\{i2pi\; s\_\{N\}\; w\_\{N$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 $w\_\{k\}$

:$q\_\{k\}\; =\; sum\_\{s\_\{k\}=-infty\}^\{infty\}\; e^\{i2pi\; s\_\{k\}\; w\_\{k$

ummary of basic protocol

The general procedure has three steps:

# Calculate the new generalized momenta $J\_\{k\}$
# Express the original Hamiltonian entirely in terms of these variables.
# Take the derivatives of the Hamiltonian with respect to these momenta to obtain the frequencies $u\_\{k\}$

Degeneracy

In some cases, the frequencies of two different generalized coordinates are identical, i.e., $u\_\{k\}\; =\; u\_\{l\}$ for $k\; eq\; l$. 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.

ee also

* Tautological one-form

References

* Lev 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