(a) In the harmonic oscillator of Exercise 1, allow all the frequencies to become equal (isotropic oscillator) so that the motion is completely degenerate. Transform to the “proper” action-angle variables, expressing the energy in terms of only one of the action variables.
(b) Solve the problem of the isotropic oscillator in action-angle variables using spherical polar coordinates. Transform again to proper action-angle variables and compare with the result of part (a). Are the two sets of proper variables the same? What are their physical significances? This problem illustrates the feasibility of separating a degenerate motion in more than one set of coordinates. The nondegenerate oscillator can be separated only in Cartesian coordinates, not in polar coordinates.
Exercise 1
Find the frequencies of a three-dimensional harmonic oscillator with unequal force constants using the method of action-angle variables. Obtain the solution for each Cartesian coordinate and conjugate momentum as functions of the action-angle variables.
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