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The harmonic oscillator

  1. Apr 5, 2013 #1
    Hi all,

    this is my first time on PF.

    I do not know English but I have a problem of a harmonic oscillator.
    I have rather large head, help me please , I do not know what else to do ...
    I have this problem:

    Consider the harmonic oscillator with an additional repulsive
    cubic force, whose potential is U(q1)=[itex]\frac{k}{2}[/itex]*[itex]q1^{2}[/itex] - k'[itex]q1^{4}[/itex], (k, k > 0), and study all
    possible solutions, periodic and non-periodic.

    I do know the Hamiltonian and the equation solution of the system, giving

    q1*=[itex]\sqrt{1/2}[/itex][itex]\int(dq1/\sqrt{(E/m)-U(q1)})[/itex]

    I tried to do it by trigonometric substitution but does not work, i do not know if anyone could give me some idea of how I can solve, I'll be very grateful.
     
  2. jcsd
  3. Apr 6, 2013 #2

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    Hi. Nobody else has responded to your question, so I guess I'll give it a try.

    The equation that I think you meant is this:

    [itex]t = \sqrt{m/2}\int(dq1/\sqrt{E-U(q1)})[/itex]

    If we change to new variables [itex]x[/itex] and [itex]s[/itex] where
    [itex]x = A q1[/itex] and [itex]s = B t[/itex], where [itex]A[/itex] and
    [itex]B[/itex] are constants, we can choose the constants to make the
    equation look like this:

    [itex]t = \int(dx/\sqrt{(1-x^2)(1- \lambda^2 x^2)}[/itex]

    where [itex]\lambda[/itex] is yet another constant.

    That's called the "Jacobi form of the incomplete elliptic integral of the first kind", [itex]F(s;\lambda)[/itex].

    http://en.wikipedia.org/wiki/Jacobi..._of_nonlinear_ordinary_differential_equations
     
  4. Apr 8, 2013 #3
    Thank you.. :D
     
    Last edited: Apr 8, 2013
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