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

[alex.dranoel]

Homework Statement

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
$$ U = U_0 \tan^2\alpha x.$$

Homework 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.

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

 

[alex.dranoel]

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 :)