Action-angle Variables
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Action-angle Variables

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 parameterize 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 ${\displaystyle W(\mathbf {q} )}$ (not Hamilton's principal function ${\displaystyle S}$). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian ${\displaystyle K(\mathbf {w} ,\mathbf {J} )}$ is merely the old Hamiltonian ${\displaystyle H(\mathbf {q} ,\mathbf {p} )}$ expressed in terms of the new canonical coordinates, which we denote as ${\displaystyle \mathbf {w} }$ (the action angles, which are the generalized coordinates) and their new generalized momenta ${\displaystyle \mathbf {J} }$. We will not need to solve here for the generating function ${\displaystyle W}$ itself; instead, we will use it merely as a vehicle for relating the new and old canonical coordinates.

Rather than defining the action angles ${\displaystyle \mathbf {w} }$ directly, we define instead their generalized momenta, which resemble the classical action for each original generalized coordinate

${\displaystyle J_{k}\equiv \oint p_{k}\,\mathrm {d} q_{k}}$

where the integration path is implicitly given by the constant energy function ${\displaystyle E=E(q_{k},p_{k})}$. Since the actual motion is not involved in this integration, these generalized momenta ${\displaystyle J_{k}}$ are constants of the motion, implying that the transformed Hamiltonian ${\displaystyle K}$ does not depend on the conjugate generalized coordinates ${\displaystyle w_{k}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}J_{k}=0={\frac {\partial K}{\partial w_{k}}}}$

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

${\displaystyle w_{k}\equiv {\frac {\partial W}{\partial J_{k}}}}$

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

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

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}w_{k}={\frac {\partial K}{\partial J_{k}}}\equiv \nu _{k}(\mathbf {J} )}$

The right-hand side is a constant of the motion (since all the ${\displaystyle J}$'s are). Hence, the solution is given by

${\displaystyle w_{k}=\nu _{k}(\mathbf {J} )t+\beta _{k}}$

where ${\displaystyle \beta _{k}}$ is a constant of integration. In particular, if the original generalized coordinate undergoes an oscillation or rotation of period ${\displaystyle T}$, the corresponding action angle ${\displaystyle w_{k}}$ changes by ${\displaystyle \Delta w_{k}=\nu _{k}(\mathbf {J} )T}$.

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

${\displaystyle \Delta w_{k}\equiv \oint {\frac {\partial w_{k}}{\partial q_{k}}}\,\mathrm {d} q_{k}=\oint {\frac {\partial ^{2}W}{\partial J_{k}\,\partial q_{k}}}\,\mathrm {d} q_{k}={\frac {\mathrm {d} }{\mathrm {d} J_{k}}}\oint {\frac {\partial W}{\partial q_{k}}}\,\mathrm {d} q_{k}={\frac {\mathrm {d} }{\mathrm {d} J_{k}}}\oint p_{k}\,\mathrm {d} q_{k}={\frac {\mathrm {d} J_{k}}{\mathrm {d} J_{k}}}=1}$

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

${\displaystyle \nu _{k}(\mathbf {J} )={\frac {1}{T}}}$

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

${\displaystyle q_{k}=\sum _{s_{1}=-\infty }^{\infty }\sum _{s_{2}=-\infty }^{\infty }\cdots \sum _{s_{N}=-\infty }^{\infty }A_{s_{1},s_{2},\ldots ,s_{N}}^{k}e^{i2\pi s_{1}w_{1}}e^{i2\pi s_{2}w_{2}}\cdots e^{i2\pi s_{N}w_{N}}}$

where ${\displaystyle A_{s_{1},s_{2},\ldots ,s_{N}}^{k}}$ is the Fourier series coefficient. In most practical cases, however, an original generalized coordinate ${\displaystyle q_{k}}$ will be expressible as a Fourier series in only its own action angles ${\displaystyle w_{k}}$

${\displaystyle q_{k}=\sum _{s_{k}=-\infty }^{\infty }A_{s_{k}}^{k}e^{i2\pi s_{k}w_{k}}}$

## Summary of basic protocol

The general procedure has three steps:

1. Calculate the new generalized momenta ${\displaystyle J_{k}}$
2. Express the original Hamiltonian entirely in terms of these variables.
3. Take the derivatives of the Hamiltonian with respect to these momenta to obtain the frequencies ${\displaystyle \nu _{k}}$

## Degeneracy

In some cases, the frequencies of two different generalized coordinates are identical, i.e., ${\displaystyle \nu _{k}=\nu _{l}}$ for ${\displaystyle k\neq l}$. 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.

## References

• 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).
• H. Goldstein, (1980) Classical Mechanics, 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9
• G. Sardanashvily, (2015) Handbook of Integrable Hamiltonian Systems, URSS. ISBN 978-5-396-00687-4
• Previato, Emma (2003), Dictionary of Applied Math for Engineers and Scientists, CRC Press, Bibcode:2003dame.book.....P, ISBN 978-1-58488-053-0