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 parameterize the coordinates on the torus. 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. Action angles result from a type-2 canonical transformation where the generating function is Hamilton's characteristic function W ( q ) {displaystyle W(mathbf {q} )} (not Hamilton's principal function S {displaystyle S} ). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian K ( w , J ) {displaystyle K(mathbf {w} ,mathbf {J} )} is merely the old Hamiltonian H ( q , p ) {displaystyle H(mathbf {q} ,mathbf {p} )} expressed in terms of the new canonical coordinates, which we denote as w {displaystyle mathbf {w} } (the action angles, which are the generalized coordinates) and their new generalized momenta J {displaystyle mathbf {J} } . We will not need to solve here for the generating function W {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 w {displaystyle mathbf {w} } directly, we define instead their generalized momenta, which resemble the classical action for each original generalized coordinate where the integration path is implicitly given by the constant energy function E = E ( q k , p k ) {displaystyle E=E(q_{k},p_{k})} . Since the actual motion is not involved in this integration, these generalized momenta J k {displaystyle J_{k}} are constants of the motion, implying that the transformed Hamiltonian K {displaystyle K} does not depend on the conjugate generalized coordinates w k {displaystyle w_{k}} where the w k {displaystyle w_{k}} are given by the typical equation for a type-2 canonical transformation Hence, the new Hamiltonian K = K ( J ) {displaystyle K=K(mathbf {J} )} depends only on the new generalized momenta J {displaystyle mathbf {J} } .