Feynman's bridge between the Kepler's laws (in particular, the relationship between the period of an orbit and the length of the major axis) and Newton's law of gravitation (that the gravitational force goes inversely as the square of the distance) is very terse. He's not proving that these two things are equivalent - only that the they are equivalent for circular orbits.
"The reason for that, is something like this...."
[more ]The first demonstration that we need is that Newton's Laws of Motion are equivalent to Kepler's observation that a planet sweeps equal areas in equal times. To be more precise, what we are demonstrating is that a body constrained by Newton's laws would necessarily sweep equal areas in equal times.
I've been studying Feynman's Lost Lecture on and off for years now, trying to come to terms with the nagging feeling that something was missing.
So as I work thought the demonstration yet again, I'm trying to fill the gaps.
My first problem was actually understanding what he was claiming to demonstrate - from the language he uses, it sounds as though it is, in total, much more grand and complete than what he delivers, and the delivery is sufficiently compelling that upon reaching the end, you may not notice the lot.
The overall pattern of the demonstration works like this.
First, Feynman repeats Newton's demonstration that equal areas in equal times is equivalent to Newton's first two laws - undisturbed motion is constant, changes in motion are in the direction of the disturbance.
Next, Feynman does a quick hand wave of the equivalence of Kepler's observation of the period of an orbit with Newton's law of gravitation. Feynman is a bit quick through here, so you may miss the scope of the observation - he moves from the period of circles to the inverse square law, and then stops. We never return to demonstrate that the relationship holds in the general case (ellipses). It does, of course, but it's not relevant to Feynman's demonstration.
Now the true demonstration begins, as "cooked up" by Feynman. First comes the transformation of equal areas in equal times to equal changes in velocity in equal orbital angles. As the text points out, he's a bit loose with his triangles - but it's all right; presenting the rigor doesn't make things more right.
Equal changes in equal angles does produce a polygon, and Archimedes demonstrated long ago that regular regular polygon becomes a circle as we make the angles infinitesmal. Since the changes in velocity are always directed toward the sun, it must be represented by the center of this circle, but the velocities themselves radiate from an off-center point.
Feynman now rotates the velocity diagram to get everything to line up, and we have almost everything that we need - the velocity is given by the length of the arrow on our diagram, the direction is perpendicular to the arrow (because we turned the diagram around), the angle in the orbit must match the angle from the sun to the head of the arrow, and the distance must be....
Whoops - Feynman says "take the perpendicular bisector" without ever justifying the choice of that particular perpendicular. As it happens, you can demonstrate that it is necessary from equal areas in equal times.
At this point, where are we? We have three laws from Newton. From these we can demonstrate easily enough that Kepler's observation of equal areas in equal times, and as Feynman shows we can produce from these three laws a demonstration that closed orbits solutions must be ellipses. It remains to be shown that these solutions have the right period, and in particular that the period is a function of the length of the major axis, but not a function of the eccentricity.
Before then moving on to Rutherford scattering, Feynman tosses off two additional observations: that you can use the same kind of velocity diagram to produce parabolic orbits and hyperbolic orbits. Most unsatisfactory - if you place the second focus on the velocity circle, then the perpendicular bisectors we had been using now all go through the sun. That Can't Be Right [tm]. After all, the magic of a parabolic mirror is to but the light source at the focus - not to embed it within the mirror itself.