# Central potential problem

Gold Member

## Homework Statement

A particle of mass m is under a central potential of the form $U(r)=-\frac{\alpha }{r^2}$ where alpha is a positive constant.
At time t=0, the spherical coordinates of the particle are worth $r=r_0$, $\theta = \pi /2$ and $\phi=0$. The corresponding time derivatives are given by $\dot r <0$, $\dot \theta =0$ and $\dot \phi \neq 0$.
The total energy is 0 and the modulus of the angular momentum is worth $\sqrt {m \alpha }$.
1)Write down the Lagrangian of the particule.
2)Find $r(t)$ and $\dot r (t)$ expressed in terms of m, alpha and $r_0$.
3)Same as in 2) but with phi(t) and $\dot \phi (t)$ and find the trajectory $r(\phi )$.
4)Calculate the time in which the particle reach the origin of the coordinate system. How many orbits does it describes before reaching it?

L=T-V.
E=T+V.

## The Attempt at a Solution

I've made a sketch. Since theta is constant and $\theta =\pi/2$, the motion is constrained into the xy plane. Therefore the angular momentum is with respect to $\phi$, namely it is worth $P_\phi = \frac{\partial L }{\partial \dot \phi}$ where L is the Lagrangian.
In spherical coordinates, $T=\frac{m}{2}(\dot r^2+r^2 \dot \theta ^2 \sin \phi + r^2\dot \phi ^2)$. But here $\dot \theta =0$. So that $p_ \phi =mr^2 \dot \phi$. I am told that $|r^2 \dot \phi |=\sqrt {\frac{\alpha }{m}}$.
1)So that the Lagrangian reduces to $L=\frac{m}{2}(\dot r ^2 + \sqrt {\frac{\alpha }{m}} \dot \phi )+\frac{\alpha }{r^2}$.
I still didn't use the fact that the total energy vanishes...
2)Euler-Lagrange equation for r gives me $\ddot r +\frac{\alpha }{m r^3}=0$. I don't know how to solve this DE. Since $\dot r$ does not appear I think the substitution $v=\dot r$ should work, but I don't reach anything with it.
So I'm basically stuck here and I'm wondering whether I'm over complicating stuff because I'm not using the fact that $\dot r <0$ and $E=0$.
Any help is greatly appreciated.

Multiply both sides with $\dot{r}$ and integrate wrt t to get a first order equation.