I haven't read the historical development, but this is my guess:Originally posted by StephenPrivitera
Is it derived from the fact that F=ma and planets travel in ellipses?
That works too... actually, I don't know for sure which (if any) is the method Newton used.EDIT:
This site used Kepler's Laws to derives Newton's
http://www.physics.ubc.ca/~outreach/phys420/p420_95/tracy/universal.html [/B]
I guess I am not aware of Galileo's work.Originally posted by Ambitwistor
Given Galileo's work, Newton knew that the gravitational force on a body had to be proportional to its mass.
Do you happen to know how to do this? I can't seem to figure it out for myself.Originally posted by Ambitwistor
He then calculated the orbits that would result from such a force law
Yes.Originally posted by StephenPrivitera
The graph of the gravitational potential U=-GMm/r looks like a hyperbola to me. If E<0, then certainly the particle is bounded in that it will not reach infinity.
It's hard to recover the shape of the orbit just by looking at the shape of the potential. You have to actually solve the equations of motion.
But what does this say about the shape of the orbit?
To see that, you have to work with the effective potential, with the 1/r^{3} "centrifugal barrier" term with angular momentum.
Also, the fact that E<0 does not put a lower bound on r (except of course r>0). What about the graph indicates that the U of the particle will oscillate?