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Integration: Action Variable J

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data
    A particle of mass m moves in one dimension subject to the potential

    [tex] V = \frac{a}{\sin^2(x/x_0)} [/tex]

    Obtain an integral expression for Hamilton's characteristic function. Under what conditions can action-angle variables be used? Assuming these are met, find the frequency of oscillation by the action-angle method. (The integral for J can be evaluated by manipulating the integrand so that the square root appears in the denominator.)

    2. The relevant equations

    I set up a Hamiltonian

    [tex] H = \frac{p^2}{2m} + \frac{a}{\sin^2(x/x_0)} [/tex].

    Since it doesn't depend on time explicitly, then I can express Hamilton's principal function S as

    [tex] S = W(x, \alpha) + \alpha t [/tex].

    The Hamilton-Jacobi equation is then given by

    [tex] \frac{1}{2m} \left( \frac{\partial W}{\partial x} \right)^2 + \frac{a}{\sin^2(x/x_0)} = \alpha [/tex]

    3. The attempt at a solution

    a.) Obtain an integral solution for Hamilton's characteristic function.

    Isolating the partial derivative of the characteristic function W, I get an integral expression

    [tex]W = \int \sqrt{2m\alpha - \frac{2ma}{\sin^2(x/x_0)}} dx [/tex].

    b.) Under what conditions can action-angle variables be used?

    The condition that must be met is [tex] \alpha \geq \frac{a}{\sin^2(x/x_0)}} [/tex].

    c.) Assuming these are met, find the frequency of oscillation by the action-angle method. (The integral for J can be evaluated by manipulating the integrand so that the square root appears in the denominator.)

    For finding the frequency of oscillation, I'm having a problem with integrating the action variable J:

    [tex] J = \int p dx = \int \sqrt{2m\alpha - \frac{2ma}{\sin^2(x/x_0)}} dx [/tex].

    I tried to follow the hint, so I got

    [tex] J = \int \frac{2m\alpha - \frac{2ma}{\sin^2(x/x_0)}}{\sqrt{2m\alpha - \frac{2ma}{\sin^2(x/x_0)}}} dx [/tex].

    Still, I'm stuck. How do I integrate this? Thanks!
     
  2. jcsd
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