What is the significance of potential energy function in central force fields?

[benitta]

hi guys could some one explain what this means

when a particle moves in an attractive central force field with a potential energy function V(x) = -k/r, for a gives non-zero value of angular momentum, at a certain time there is a minimum energy for which,it is possible to find the solutions to the equations of motions. at this minimum energy the particle moves in an elliptical orbit.

 

[Galileo]

It's not the most lucid description I've ever seen, but what I think it means to say is that, for a particle in an inverse-square force field (I`ll take a star and a planet), the trajectory of the planet is an ellipse under a few assumptions:
- The planet does not head straight towards the sun (non-zero angular momentum)
- The total energy of the particle must be negative.

If the total energy is positive it will not get caught into an orbit but can reach infinity (where V=0) with kinetic energy left. The trajectory will be a hyperbola.
If the total energy is (exactly) zero you have a special case and the trajectory is a parabola.