There's some fascinating history behind Isaac Newton's theory of gravity, which ultimately also documents his laws of motion. Here is a primer I'm taking from a different post I wrote for another thread (this post does not touch on the calculus part of the OP, just the gravity part):
Below is
Isaac Newton's vis viva equation (My goodness, this equation is forcefully alive!
):
v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right)
where,
r is the distance from the smaller, orbiting body to the larger body,
a is the semi-major axis of the ellipse traced out by the smaller, orbiting body.
GM is the standard gravitational parameter of the larger body (which is Newton's gravitational constant times the mass of the larger body).
and v is the velocity of the smaller body at the given position of its orbit (its speed for a given value of r).
From a historical point of view, the relationship embodied in his
vis viva equation changed the world forever after, at least indirectly if not more.
You might not be familiar with this equation, and perhaps have never even heard of it. It's not the equation for which Newton is best known. But from a historical point of view Newton's
vis viva relationship is paramount to a revolutionary change. The
vis viva relationship is the first to show
how a force could cause a moon/planet to orbit in a ellipse. But historically, it was involved in much more.
It was on an August day in 1684 when Edmond Halley (eventually of Halley's comet fame), paid a call on a young Issac Newton. Halley and Newton had already met some years before. Newton had made a small name for himself by his studies with light and inventing the reflecting telescope. The purpose of Halley's visit this time was to inquire if Newton had made progress with the problem of how a gravitational force might cause an orbiting body to move in an ellipse around the heavier, more massive body.
Both Newton and Halley knew that planets move in ellipses. This had been known for more than a half century by the work of Johannes Kepler's
Astronomia nova. When traveling to visit Newton, Halley might have even suspected that the gravitational force obeys the inverse square law (although that's difficult to say for sure). Inverse square law or not, Halley himself failed to show how a force (of whatever sort) would cause a planet to move around the Sun in an ellipse. Hence the reason for calling on Newton.
So when Halley arrived in Cambridge that day in 1648, he asked Newton about it. Newton said that he had already solved it, and that the result was laying around in his notes somewhere. Newton couldn't find the relationship in his notes, that day, but said he'd forward them on when he found them.
In November of that year, Newton did forward the relationship and related material (either reproducing or re-deriving them) to Halley in
De motu corporum in gyrum ("On the motion of bodies in an orbit"). Halley was impressed to say the least, and urged Newton to create a more detailed version, assuring him it would be published by the Royal Society.
Isaac Newton followed Halley's advice and created the much more detailed
Philosophiæ Naturalis Principia Mathematica. This work, I'm sure you've heard of. It contains Newton's laws of motion, his universal law of gravity (including the
vis viva equation, along with the inverse square law and the rest), and is the very basis of Newtonian physics that we still use today.
The story doesn't simply end with the publication. The Royal Society was broke at the time, and did not have the money to publish Newton's masterpiece. Halley himself had to raise most of the publication expenses, much of them coming out of his own pocket. Apparently, the Royal Society even offered to compensate Halley in the form of unsold copies of
fish books, "The History of the Fish."
But it all started with Edmund Halley's short visit to Isaac Newton that one August day, looking for something along the lines of what we now call the
vis viva equation.
I won't derive the
vis viva equation here, since there is already a good derivation on the Wiki site:
http://en.wikipedia.org/wiki/Vis-viva_equation.
