Kepler’s first law states an object orbits another object under a gravitational pull in a conic section orbit. If the orbit is closed, it is an ellipse. It also turns out to be really hard to prove. You either have to use calculus and differential equations, or use geometry with lines and stuff and know well the properties of the ellipse. Either way, you have to set up the problems in creative ways. In this post, I would like to collect all the ways Kepler’s first law can be derived. I have always found it annoying how scattered the proofs were, and I would like to leave this behind for anyone who is itching to find out how Newton’s laws implies elliptical orbit and vice versa.
For a newbie, the best proof out there is in my opinion Feynman’s geometric proof. While it is still complicated, it is not as hard as the other proofs to understand. You don’t have to know anything too complicated except for some of the properties of the ellipse, and get used to the methods of geometry. It is also great for its clarity, unlike Newton’s geometric proof. Newton’s proof is convoluted and use really complicated geometry, but if you would like to know how the master himself did it, there it is.
The most common derivation is the differential equation approach. It is the standard textbook approach, and if you know something about calculus and differential equations, it is easier to swallow. Then there is the more complicated version which takes account of the fact that two move around a center of mass, instead of one around the other.
My favorite version, though, is the one that uses the Laplace-Runge-Lenz vector. Its derivation is elegantly simple, following directly from the approach. In the other differential equation method, you have to find the acceleration in terms of polar coordinates and then do a creative substitution that makes them end up as a simple second order differential equation. This one is somewhat less convoluted than that, and once you get the vector, you are only one step away from Kepler’s first law. In The Mechanical Universe and Beyond, video 22 titled The Kepler Problem uses this derivation.
Finally, I know there is the one that uses the complex function. Unfortunately, I can’t find it online. It is contained in this book, though.
If anyone knows of other alternatives, I can post it here.