Mechanics - particle in a potential

[Pagan Harpoon]

Homework Statement

Find the equation of motion for a particle moving in a potential $$U(x)=Vtan^2(ax)$$ with V>0. The motion occurs in one dimension.

Homework Equations

$$\frac{\partial L}{\partial x}=\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}$$ (*)

The Attempt at a Solution

$$L=\frac{1}{2} m\dot{x}^2-Vtan^2(ax)$$

By taking derivatives of this and applying (*), it is easy to arrive at the differential equation:

$$\ddot{x}=-\frac{2Va}{m} \frac{sin(ax)}{cos^3(ax)}$$

but this isn't much use because there's no way I can solve that.

So another approach I tried is this:

$$L=E_k-U(x)=(E-U(x))-U(x)=E-2U(x)$$

Where E is constant.

Now apparently $$\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}=0$$ because $$\dot{x}$$ doesn't appear in L. So,

$$0=\frac{\partial L}{\partial x}=-\frac{2Va}{m} \frac{sin(ax)}{cos^3(ax)}$$

But this can't be right, because it implies that x=n/aPi where n is an integer. Clearly, x should be a function of time, x_0 and v_0 and not a constant. However, I can't identify what is wrong with the analysis I did.

Thank you.

 

[Pagan Harpoon]

No ideas?

 

[ehild]

There is potential, so the force is conservative. The energy is conserved, T+V = E. You can get the velocity, v= dx/dt in terms of x: dx/dt = f(x). You can solve this differential equation in principle, to get the equation of motion in terms of the time.

ehild