- 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_{Nwhere 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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