Effect of Additional 1/r^2 Potential Energy on Elliptical Motion

[LionMagnus]

Homework Statement

Hi, this is one the problem on my homework, but I don't know where t start with this problem. We have equation of a particle's potential energy field:
U(r)=-GMm/r+Cm/r^2 C is just a constant. I need to solve for its radial motion and angular motion, then if the motion of the particle is elliptical, what will be the effect of the addtional 1/r^2 potential energy term? Any amount of help will be appreciated!

Homework Equations

U(r)=-GMm/r+Cm/r^2 C

The Attempt at a Solution

No attempt yet, don't quite know where to start

 

[tman12321]

The potential you have written down has a 1/r term and a 1/r^2 term. The first is the gravitational potential, and the second has the same effect as a centrifugal force, and will simply add to that term. The standard way of solving the equation of orbit involves a few tricks. First define u = 1/r. The rewrite d/dt in terms of some constant multiplied by an azimuthal derivative, say d/dθ. Finally use the Euler-Lagrange equation to identify constants of motion (hint: angular momentum is conserved, this will give you the azimuthal motion). Rewrite the radial equation of motion in terms of all these new variables. Now you have a standard differential equation for which the answer is know. I don't know if you're really expected to go through all this (it should be in your textbook) but the upshot of it all is that r(θ) = a/[1+εcos(θ)], where a is some constant, and ε is the eccentricity, both constants of integration determined by initial conditions. The magnitude of ε (whether it is between 0 and 1, or greater than 1) will determine if you have a bounded or unbounded orbit. This should get you started. I would strongly recommend reading up on this material in your textbook.