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| In [[classical mechanics]], '''action-angle coordinates''' are a set of [[canonical coordinates]] useful in solving many [[integrable system]]s. The method of action-angles is useful for obtaining the [[frequency|frequencies]] of oscillatory or rotational motion without solving the [[equations of motion]]. Action-angle coordinates are chiefly used when the [[Hamilton–Jacobi equation]]s are completely separable. (Hence, the [[Hamiltonian (quantum mechanics)|Hamiltonian]] does not depend explicitly on time, i.e., the [[conservation of energy|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.
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| 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, [[Albert Einstein|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.
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| Action-angle coordinates are also useful in [[perturbation theory]] of [[Hamiltonian mechanics]], especially in determining [[adiabatic invariant]]s. 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.
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| 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.
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| ==Derivation==
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| Action angles result from a type-2 [[canonical transformation]] where the generating function is [[Hamilton–Jacobi equation|Hamilton's characteristic function]] <math>W(\mathbf{q})</math> (''not'' Hamilton's principal function <math>S</math>). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian <math>K(\mathbf{w}, \mathbf{J})</math> is merely the old Hamiltonian <math>H(\mathbf{q}, \mathbf{p})</math> expressed in terms of the new [[canonical coordinates]], which we denote as <math>\mathbf{w}</math> (the '''action angles''', which are the [[generalized coordinates]]) and their new generalized momenta <math>\mathbf{J}</math>. We will not need to solve here for the generating function <math>W</math> itself; instead, we will use it merely as a vehicle for relating the new and old [[canonical coordinates]].
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| Rather than defining the action angles <math>\mathbf{w}</math> directly, we define instead their generalized momenta, which resemble the [[action (physics)|classical action]] for each original [[generalized coordinate]]
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| :<math>
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| J_{k} \equiv \oint p_{k} dq_{k}
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| </math>
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| where the integration path is implicitly given by the constant energy function <math>E=E(q_{k},p_{k})</math>. Since the actual motion is not involved in this integration, these generalized momenta <math>J_{k}</math> are constants of the motion, implying that the transformed Hamiltonian <math>K</math> does not depend on the conjugate [[generalized coordinates]] <math>w_{k}</math> | |
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| :<math>
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| \frac{d}{dt} J_{k} = 0 = \frac{\partial K}{\partial w_{k}}
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| </math>
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| where the <math>w_{k}</math> are given by the typical equation for a type-2 [[canonical transformation]]
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| :<math>
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| w_{k} \equiv \frac{\partial W}{\partial J_{k}}
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| </math>
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| Hence, the new Hamiltonian <math>K=K(\mathbf{J})</math> depends only on the new generalized momenta <math>\mathbf{J}</math>.
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| The dynamics of the action angles is given by [[Hamilton's equations]]
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| :<math>
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| \frac{d}{dt} w_{k} = \frac{\partial K}{\partial J_{k}} \equiv \nu_{k}(\mathbf{J})
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| </math>
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| The right-hand side is a constant of the motion (since all the <math>J</math>'s are). Hence, the solution is given by
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| :<math> | |
| w_{k} = \nu_{k}(\mathbf{J}) t + \beta_{k}
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| </math>
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| where <math>\beta_{k}</math> is a constant of integration. In particular, if the original [[generalized coordinate]] undergoes an oscillation or rotation of period <math>T</math>, the corresponding action angle <math>w_{k}</math> changes by <math>\Delta w_{k} = \nu_{k}(\mathbf{J}) T</math>.
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| These <math>\nu_{k}(\mathbf{J})</math> are the frequencies of oscillation/rotation for the original [[generalized coordinate]]s <math>q_{k}</math>. To show this, we integrate the net change in the action angle <math>w_{k}</math> over exactly one complete variation (i.e., oscillation or rotation) of its [[generalized coordinate]]s <math>q_{k}</math>
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| :<math> | |
| \Delta w_{k} \equiv \oint \frac{\partial w_{k}}{\partial q_{k}} dq_{k} =
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| \oint \frac{\partial^{2} W}{\partial J_{k} \partial q_{k}} dq_{k} =
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| \frac{d}{dJ_{k}} \oint \frac{\partial W}{\partial q_{k}} dq_{k} =
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| \frac{d}{dJ_{k}} \oint p_{k} dq_{k} = \frac{dJ_{k}}{dJ_{k}} = 1
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| </math>
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| Setting the two expressions for <math>\Delta w_{k}</math> equal, we obtain the desired equation
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| :<math>
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| \nu_{k}(\mathbf{J}) = \frac{1}{T}
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| </math>
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| The action angles <math>\mathbf{w}</math> are an independent set of [[generalized coordinates]]. Thus, in the general case, each original generalized coordinate <math>q_{k}</math> can be expressed as a [[Fourier series]] in ''all'' the action angles
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| :<math>
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| 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^{i2\pi s_1 w_1} e^{i2\pi s_2 w_2} \ldots e^{i2\pi s_N w_N}
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| </math>
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| where <math>A^{k}_{s_{1}, s_{2}, \ldots, s_{N}}</math> is the Fourier series coefficient. In most practical cases, however, an original generalized coordinate <math>q_{k}</math> will be expressible as a [[Fourier series]] in only its own action angles <math>w_{k}</math>
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| :<math>
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| q_{k} = \sum_{s_{k}=-\infty}^\infty e^{i2\pi s_{k} w_{k}}
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| </math>
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| ==Summary of basic protocol==
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| The general procedure has three steps:
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| # Calculate the new generalized momenta <math>J_{k}</math>
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| # Express the original Hamiltonian entirely in terms of these variables.
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| # Take the derivatives of the Hamiltonian with respect to these momenta to obtain the frequencies <math>\nu_{k}</math>
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| ==Degeneracy==
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| In some cases, the frequencies of two different [[generalized coordinate]]s are identical, i.e., <math>\nu_{k} = \nu_{l}</math> for <math>k \neq l</math>. In such cases, the motion is called '''degenerate'''.
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| 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]].
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| 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 both [[spherical coordinates]] and [[parabolic coordinates]].
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| ==See also==
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| * [[Tautological one-form]]
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| * [[Integrable system]]
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| * [[Superintegrable Hamiltonian system]]
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| ==References==
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| * 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).
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| * H. Goldstein, (1980) ''Classical Mechanics'', 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9
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| [[Category:Hamiltonian mechanics]]
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