Find the one dimensional particle motion in a given potential

[Fallen Seraph]

I'm not looking for a solution, but rather trying to understand the question.

We've been given a series of potentials, U(x), and have been told to find the one-dimensional particle motion in them. For example:


U(x) = V(tan^2(cx)), V>0

My initial reaction was just to solve it for x(t), but after having found a(x), I'm not so sure that this is possible analytically... (I can't visualise a solution to -ma=2cV(tan(cx))(1+tan^2(cx))

So perhaps the question is asking to find the period of the oscillatory motion? But it certainly doesn't look like that's what it's asking...

 

[muppet]

Well I've tried to think of a nice trick using the chain rule or similar, but can't solve $$-\frac{1}{m}\frac{d}{dt}x(t)= U(x(t))$$. Could it be you just have to qualitatively describe the resultant motion, or will all of your potentials result in oscillation?

 

[Astronuc]

Well U(x) = V(tan²(cx)), V>0, has an equilibrium point about x = 0.

tan² x is an even function about x = 0.

 

[Sourabh N]

since 'a' is a function of x, you can use a = vdv/dx to get v as function of x and then v = dx/dt.