# Action-angle coordinates

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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 $W\left(mathbf\left\{q\right\}\right)$ ("not" Hamilton's principal function $S$). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian $K\left(mathbf\left\{w\right\}, mathbf\left\{J\right\}\right)$ is merely the old Hamiltonian $H\left(mathbf\left\{q\right\}, mathbf\left\{p\right\}\right)$ expressed in terms of the new canonical coordinates, which we denote as $mathbf\left\{w\right\}$ (the action angles, which are the generalized coordinates) and their new generalized momenta $mathbf\left\{J\right\}$. 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 $mathbf\left\{w\right\}$ directly, we define instead their generalized momenta, which resemble the classical action for each original generalized coordinate

:$J_\left\{k\right\} equiv oint p_\left\{k\right\} dq_\left\{k\right\}$

where the integration is over all possible values of $q_\left\{k\right\}$, given the energy $E$. Since the actual motion is not involved in this integration, these generalized momenta $J_\left\{k\right\}$ are constants of the motion, implying that the transformed Hamiltonian $K$ does not depend on the conjugate generalized coordinates $w_\left\{k\right\}$

:$frac\left\{d\right\}\left\{dt\right\} J_\left\{k\right\} = 0 = frac\left\{partial K\right\}\left\{partial w_\left\{k$

where the $w_\left\{k\right\}$ are given by the typical equation for a type-2 canonical transformation

:$w_\left\{k\right\} equiv frac\left\{partial W\right\}\left\{partial J_\left\{k$

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

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

:$frac\left\{d\right\}\left\{dt\right\} w_\left\{k\right\} = frac\left\{partial K\right\}\left\{partial J_\left\{k equiv u_\left\{k\right\}\left(mathbf\left\{J\right\}\right)$

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 $w_\left\{k\right\}$ changes by $Delta w_\left\{k\right\} = u_\left\{k\right\}\left(mathbf\left\{J\right\}\right) T$.

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

:$Delta w_\left\{k\right\} equiv oint frac\left\{partial w_\left\{k\left\{partial q_\left\{k dq_\left\{k\right\} = oint frac\left\{partial^\left\{2\right\} W\right\}\left\{partial J_\left\{k\right\} partial q_\left\{k dq_\left\{k\right\} = frac\left\{d\right\}\left\{dJ_\left\{k oint frac\left\{partial W\right\}\left\{partial q_\left\{k dq_\left\{k\right\} = frac\left\{d\right\}\left\{dJ_\left\{k oint p_\left\{k\right\} dq_\left\{k\right\} = frac\left\{dJ_\left\{k\left\{dJ_\left\{k = 1$

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

:$u_\left\{k\right\}\left(mathbf\left\{J\right\}\right) = frac\left\{1\right\}\left\{T\right\}$

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

:$q_\left\{k\right\} = sum_\left\{s_\left\{1\right\}=-infty\right\}^\left\{infty\right\} sum_\left\{s_\left\{2\right\}=-infty\right\}^\left\{infty\right\} ldots sum_\left\{s_\left\{N\right\}=-infty\right\}^\left\{infty\right\} A^\left\{k\right\}_\left\{s_\left\{1\right\}, s_\left\{2\right\}, ldots, s_\left\{N e^\left\{i2pi s_\left\{1\right\} w_\left\{1 e^\left\{i2pi s_\left\{2\right\} w_\left\{2 ldots e^\left\{i2pi s_\left\{N\right\} w_\left\{N$where $A^\left\{k\right\}_\left\{s_\left\{1\right\}, s_\left\{2\right\}, ldots, s_\left\{N$ is the Fourier series coefficient. In most practical cases, however, an original generalized coordinate $q_\left\{k\right\}$ will be expressible as a Fourier series in only its own action angles $w_\left\{k\right\}$

:$q_\left\{k\right\} = sum_\left\{s_\left\{k\right\}=-infty\right\}^\left\{infty\right\} e^\left\{i2pi s_\left\{k\right\} w_\left\{k$

ummary of basic protocol

The general procedure has three steps:

# Calculate the new generalized momenta $J_\left\{k\right\}$
# 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 $u_\left\{k\right\}$

Degeneracy

In some cases, the frequencies of two different generalized coordinates are identical, i.e., $u_\left\{k\right\} = u_\left\{l\right\}$ 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

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