The oscillation of a particle in a special potential field

[Peter Jones]

Homework Statement

Prove that the particle moves periodically between two points and find the period of small angle oscillations

Relevant Equations

The question is:
A particle of mass m moves with energy E0 and has the potential energy Ụ(x)=U0tan^2(x/a) where U0 and a are constants with units of energy and length, respectively. Prove that the particle moves periodically between two points x1, x2 which are the roots of the following equation E-m/2(v0y^2+v0z^2)-U0tan^2(x/a)=0 (v0y,v0z are the particle’s initial velocity in 0y, Oz). Find the period of small angle oscillations about all stable equilibrium points.

I couldn't prove the first one but i tried to find the period

F = -dU / dx

= - d( U0tan^2( x / a ) ) / dx

= - U0 ( ( 2 sec^2( x / a ) tan( x / a ) / a )
with F=d^2x/dt^2, tan(x/a)=x/a we have
d^2x/dt^2 + U0 ( ( 2 sec^2( x / a ) ( x / a^2 ) =0
from there i don't know how to handle the sec^2(x/a)

 

[PeroK]

Have you seen the general idea of using a Taylor series expansion of the potential function for small oscillations?

 

[PeroK]

PS For the first part, you could think about the fact that the particle cannot keep moving in the x-direction, so it must reach a turning point where $\frac{dx}{dt} = 0$.