Action-angle coordinates

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${\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}$ Second law of motion
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In classical mechanics, action-angle variables are a set of canonical coordinates that are useful in characterizing the nature of commuting flows in integrable systems when the conserved energy level set is compact, and the commuting flows are complete. Action-angle variables are also important in obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. They only exist, providing a key characterization of the dynamics, when the system is completely integrable, i.e., the number of independent Poisson commuting invariants is maximal and the conserved energy surface is compact. This is usually of practical calculational value when the Hamilton–Jacobi equation is completely separable, and the separation constants can be solved for, as functions on the phase space. Action-angle variables define a foliation by invariant Lagrangian tori because the flows induced by the Poisson commuting invariants remain within their joint level sets, while the compactness of the energy level set implies they are tori. The angle variables provide coordinates on the leaves in which the commuting flows are linear.

The connection between classical Hamiltonian systems and their quantization in the Schrödinger wave mechanics approach is made clear by viewing the Hamilton–Jacobi equation as the leading order term in the WKB asymptotic series for the Schrodinger equation. In the case of integrable systems, the Bohr–Sommerfeld quantization conditions were first used, before the advent of quantum mechanics, to compute the spectrum of the hydrogen atom. They require that the action-angle variables exist, and that they be integer multiples of the reduced Planck constant. Einstein's insight in the EBK quantization into the difficulty of quantizing non-integrable systems was based on this fact.

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 dynamical stability of integrable dynamical systems under small perturbations, is the KAM theorem, which states that the invariant tori are partially stable.

In the modern theory of integrable systems action-angle variables were used in the solution of the Toda lattice, the definition of Lax pairs, or more generally, isospectral evolution of a linear operator characterizing integrable dynamics, and interpreting the associated spectral data as action-angle variables in the Hamiltonian formulation.