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Period of a particle in a given potential (ex. from Mechanics Landau Lifshitz))

  1. Jan 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Determine the period of oscillation, as a function of the energy, when a particle of mass ##m## moves in a field for which the potential energy is
    $$ U = U_0 \tan^2\alpha x.$$


    2. Relevant equations
    The relevant equation is given by the general formula for the period ##T## of the oscillations:
    $$ T(E) = \sqrt{2m} \int_{x_1(E)}^{x_2(E)} \frac{dx}{\sqrt{E-U(x)}}$$
    where ##x_1(E)## and ##x_2(E)## are roots of ##U(x)=E##, giving the limit of the motion.


    3. The attempt at a solution
    This problem is found in Mechanics by Landau and Lifshtiz at $11 in the 3rd edition. It is basically a problem of integration. The first thing I did is to find ##x_2(E)##, which is not complicated:
    $$E = U_0 tan^2\alpha x \rightarrow x = \frac{1}{\alpha}\arctan \sqrt{\frac{E}{U_0}}$$
    Then given the symmetry of the problem, it is clear that
    $$ T(E) = 2\sqrt{2m} \int_{0}^{x_2(E)} \frac{dx}{\sqrt{E-U_0\tan^2\alpha x}}$$
    Now I am left with an integral that I didn't manage to compute while the result in the book looks very simple.

    Thanks for help
     
  2. jcsd
  3. Jan 23, 2013 #2
    I found the way to compute the integral. Firstly, subsitute ##y=\alpha x## to get ride of the ##\alpha## term and then factorize ##E## under the square root. Then substitute ##\sin z = \frac{U_0}{E}\tan y##. If you do everything correctly you will end up with the definite integral:
    $$ \int_0^{\pi/2} \frac{dz}{\frac{U_0}{E} + \sin^2 z}$$
    that you can compute by using the residue theorem.

    If any of want to try, feel free :)
     
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