Action-angle coordinates


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

: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 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_{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 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


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