equation for a type-2 canonical transformation Hence, the new Hamiltonian depends only on the new generalized momenta . The dynamics of the action angles is given by Hamilton's equations The right-hand side is a constant of the motion (since all the 's are). Hence, the solution is given by where is a constant of integration. In particular, if the original generalized coordinate undergoes an oscillation or rotation of period , the corresponding action angle changes by . These are the frequencies of oscillation/rotation for the original generalized coordinates . To show this, we integrate the net change in the action angle over exactly one complete variation (i.e., oscillation or rotation) of its generalized coordinates Setting the two expressions for equal, we obtain the desired equation The action angles are an independent set of generalized coordinates. Thus, in the general case, each original generalized coordinate can be expressed as a Fourier series in all the action angles where is the Fourier series coefficient. In most practical cases, however, an original generalized coordinate will be expressible as a Fourier series in only its own action angles Summary of basic protocol The general procedure has three steps: Calculate the new generalized momenta 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 Degeneracy In some cases, the frequencies of two different generalized coordinates are identical, i.e., for . 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. See also Integrable system Tautological one-form Superintegrable Hamiltonian system Einstein–Brillouin–Keller method References L. D. Landau and E. M. Lifshitz, (1976) Mechanics, 3rd. ed., Pergamon Press. (hardcover) and (softcover). H. Goldstein, (1980) Classical Mechanics, 2nd. ed., Addison-Wesley. G. Sardanashvily, (2015) Handbook of Integrable Hamiltonian Systems, URSS. Coordinate systems Classical mechanics Dynamical systems Hamiltonian mechanics