- Action-angle coordinates
In

classical mechanics ,**action-angle coordinates**are a set ofcanonical coordinates useful in solving manyintegrable system s. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving theequations of motion . Action-angle coordinates are chiefly used when theHamilton–Jacobi equation s are completely separable. (Hence, theHamiltonian 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 atorus , while the angle variables provide the coordinates on the torus.The

Bohr-Sommerfeld quantization conditions, used to develop quantum mechanics before the advent ofwave mechanics , state that the action must be an integral multiple ofPlanck's constant ; similarly, Einstein's insight intoEBK 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 ofHamiltonian mechanics , especially in determiningadiabatic invariant s. One of the earliest results fromchaos theory , for the non-linear perturbations of dynamical systems with a small number of degrees of freedom is theKAM 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 ofLax pairs , or more generally, the idea of theisospectral 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 newcanonical coordinates , which we denote as $mathbf\{w\}$ (the**action angles**, which are thegeneralized 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 oldcanonical 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

:$w\_\{k\}\; =\; u\_\{k\}(mathbf\{J\})\; t\; +\; eta\_\{k\}$

where $eta\_\{k\}$ 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 $Delta\; w\_\{k\}\; =\; u\_\{k\}(mathbf\{J\})\; T$.These $u\_\{k\}(mathbf\{J\})$ are the frequencies of oscillation/rotation for the original

generalized coordinate s $q\_\{k\}$. 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 itsgeneralized coordinate s $q\_\{k\}$:$Delta\; w\_\{k\}\; equiv\; oint\; frac\{partial\; w\_\{k\{partial\; q\_\{k\; dq\_\{k\}\; =\; oint\; frac\{partial^\{2\}\; W\}\{partial\; J\_\{k\}\; partial\; q\_\{k\; dq\_\{k\}\; =\; frac\{d\}\{dJ\_\{k\; oint\; frac\{partial\; W\}\{partial\; q\_\{k\; dq\_\{k\}\; =\; frac\{d\}\{dJ\_\{k\; oint\; p\_\{k\}\; dq\_\{k\}\; =\; frac\{dJ\_\{k\{dJ\_\{k\; =\; 1$

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 aFourier 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 coordinate s 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 theLaplace-Runge-Lenz vector .Degenerate motion also signals that the

Hamilton–Jacobi equation s are completely separable in more than one coordinate system; for example, the Kepler problem is completely separable in bothspherical coordinates andparabolic 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

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