Find the Time period using First Integral

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1. Jul 30, 2016

Muthumanimaran

Recently I lend the Classical mechanics book written by Goldstein from the library, In the last page, someone scribbled this problem without any solution, I am just curious and want to give a try the problem mentioned below. I just want to know whether my approach and my solution is correct or not.
1. The problem statement, all variables and given/known data

Find the Time period of the particle confined to the Potential energy of the form $V(x)=-\frac{V_{0}}{\cosh^2(\alpha{x})}$.

2. Relevant equations
$$\int_{0}^{t_0}dt=\int_{x_0}^{x}\frac{dx}{\sqrt{\frac{2}{m}(E-V(x))}}$$
Where $'E'$ is the Total Energy
and $'m'$ is the mass of the particle

3. The attempt at a solution
Since the potential energy is negative, the total Energy required to escape from the potential is zero, i.e, $E=0$,
The Potential diagram is attached with this thread, as we can see, the potential goes to 0 as $x$ goes to $\infty$
or $-\infty$, If I take one half of the Potential say from $0$ to $\infty$ and substituting in the above integral becomes,

$$\int_{0}^{\frac{T}{2}}dt=\sqrt{\frac{m}{2V_0}}\int_{0}^{\infty}cosh(\alpha{x})dx$$

Apparently the right hand side goes to infinity. And the time period of the particle confined also goes to infinity. Am I right? I solved the problem but I don't know whether my solution is correct or not. Is there any mistake in my solution?

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Last edited: Jul 30, 2016
2. Jul 30, 2016

Orodruin

Staff Emeritus
If you take E = 0 there is no classical turning point and the motion is not periodic. In order to have a periodic solution you need to assume a bound state with E < 0.