The equations of motion for have been reduced to conservation of energy,

and conservation of angular momentum,

It is straightforward to solve the second of these for ,

and substitute this into the first to obtain an equation of motion for ,

Note how the angular momentum acts like a contribution to the potential , making the area near "high potential."

In addition to finding and as functions of the time , it is also possible to find and equation relating the two,

This equation will be the starting point for our discussion of orbital motion next month