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