Solving for Particle Trajectory in Central Potential with Initial Conditions

[fluidistic]

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?

Homework Equations

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.

 

[clamtrox]

I don't know how to solve this DE.

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