Can You Solve the Keplerian Two-Body Problem in Physics?

[GaleForce]

Alright I'm really stuck on this question. I was wondering if anyone could help:

(a) Show that the total energy (per unit mass) of a particle orbiting in an attractive Keplerian potential V(r) = -GM/r is

E = (1/2)(dr/dt)^2 + (1/2)(J^2/r^2)-(GM)/r

where J = |r x v| is the particle's angular momentum (per unit mass).

(b) If the particle in part (a) has J =\= 0 and finite energy, is it possible for the particle to reach r = 0? Can the particle reach r = infinity if E < 0? If E > 0? If may help to sketch a graph of 1/2(dr/dt)^2 as a function of r for both E < 0 and E > 0. I have no idea how to even approach the problem. It really doesn't fit in with the rest of the work we've been doing so I'm completely clueless. Thanks if anyone can help out.

 

[Galileo]

a) dr/dt is only the radial part of the velocity vector \vec v. Since -GM/r is the potential energy (per unit mass), (1/2)(J^2/r^2) must be the contribution of the tangential component of the velocity to the kinetic energy p.u.m. \frac{1}{2}\vec v\cdot \vec v.

So the first thing to do is to separate the vector \vec r,\vec v into their radial and tangential components.

b) What are your thoughts on this one? What can you say if r=0?