In chapter 13 of volume 1 of The Feynman Lectures, Feynman is discussing how the work done in going around any path in a gravitational field is zero. Here's the link: http://www.feynmanlectures.caltech.edu/I_13.html
The troubling text is as follows:
'Thus we see that the work done in going along the sides of a small triangle is the same as that done going on a slant, because scosθ is equal to x. We have proved previously that the answer is zero for any path composed of a series of notches like those of Fig. 13–3, and also that we do the same work if we cut across the corners instead of going along the notches (so long as the notches are fine enough, and we can always make them very fine); therefore, the work done in going around any path in a gravitational field is zero.
This is a very remarkable result. It tells us something we did not previously know about planetary motion. It tells us that when a planet moves around the sun (without any other objects around, no other forces) it moves in such a manner that the square of the speed at any point minus some constants divided by the radius at that point is always the same at every point on the orbit. For example, the closer the planet is to the sun, the faster it is going, but by how much? By the following amount: if instead of letting the planet go around the sun, we were to change the direction (but not the magnitude) of its velocity and make it move radially, and then we let it fall from some special radius to the radius of interest, the new speed would be the same as the speed it had in the actual orbit, because this is just another example of a complicated path. So long as we come back to the same distance, the kinetic energy will be the same. So, whether the motion is the real, undisturbed one, or is changed in direction by channels, by frictionless constraints, the kinetic energy with which the planet arrives at a point will be the same.'
I don't get what quantity is the speed squared minus some constants divided by r. Is it energy? If so, how? I also don't get what corners or notches he is talking about in the first paragraph. Then, he talks about letting the planet fall from a special radius to the radius of interest. What is that supposed to mean? What does he mean by special and interest radius? How would the speed be same if the planet is moving closer to the Sun? Shouldn't it be faster?
I seem to be asking a lot of questions related to the Feynman Lectures. Thank you !