# Deriving Kepler's First Law from Newton's Law of Universal Gravitation: Interpretation of the $u = 1/r$ change of variable

I'm working through a somewhat unsatisfying derivation of Kepler's First Law from Newton's Law of Universal Gravitation. The key to obtaining the equation of motion that you can solve lies in making the change of variable $u = 1/r$ in the angular momentum of mass $m$, $L = m r^2 \omega$, which is conveniently treated as a point mass. I see the mathematical necessity of this, but what is the physical interpretation (if any)? Is this necessary because the gravitational potential of mass $m$ is $1 / r$ for large $r$?

• There is no necessity for a physical interpretation of a change of variable. Physics is invariant under changes of coordinates, the same things happen regardless of which coordinates we choose, so I don't know what you're asking for. – ACuriousMind Nov 20 '15 at 18:24
• A thorough walkthrough all the steps in the descriptions of classical $f(r)$ potentials, orbits, equations of motion, Kepler's laws and all the rest can be found in the standard amazon.com/Classical-Mechanics-Edition-Herbert-Goldstein/dp/… – gented Nov 20 '15 at 21:37
• @ACuriousMind, thank you for the reminder about invariance. Just like a wise choice of coordinate system, a similar wise choice of integration variable will ease the interpretation of your result. – Joel DeWitt Nov 20 '15 at 23:00