## Proof That Trajectory of BodyAbout Another Body is a Conic Section

Suppose a body moves under the influence of a single mass producing a gravitational field. The equation of motion of the body isTaking the cross product of this equation withgives

Nowsince angular mometum is conserved.

Hencea constant vector. We can write

Using the identitygives

Integrating this equation givesand taking the dot product withgiveswhereis the angle betweenand

Now use the identityto give

Hence