# From Newton to Kepler without infinitesimals

I've read some interesting calculus-free proofs of at least parts of the derivation of Kepler's Laws from Newton's gravitational force. One is of course Feyman's "Lost Lecture" (which was already featured here in a comprehensive post), which dervies Kepler's first law from a central inverse-square law; then there's a partial derivation using Mamikon's Theorem (MTK) of Kepler's second law from just the centrality of gravitational force.

Both of these eschew modern calculus, but they do use infinitesimals: Feynman's one uses "very small" $\Delta\theta$'s, while MTK "hides" infinitesimals within the assumption that areal velocity is "better understood" as angular momentum.

My questions are: assuming concepts such as velocity and acceleration as primitive, is there in classical mechanics a complete derivation of Kepler's Laws from Newton's inverse-square law which doesn't rely on modern calculus and infinitesimals, vanishing quantities and so on? Maybe using just indivisibles? If not - is there a way to prove that no such derivation can exist?

The proofs I mentioned above seem to hint at the possibility of the first option (MTK in particular, which basically replaces infinitesimals with indivisibles), but I'm not able to rework them in such a way that infinitesimals are totally removed (eg. to use Mamikon's Theorem to prove the relation between angular momentum and areal velocity). But maybe there is some very strong reason (ellipticity of integrals?) for which no proof based on Mamikon's Theorem or anything close to it can directly relate angular momentum and areal velocity...

I actually have no clue. Does any of you?

• Not sure what the question here is. Of course there is a path through classical mechanics (and in particular to Kepler's laws) without "infinitesimals" - use standard calculus! – ACuriousMind Nov 1 '15 at 16:00
• Yes, sure. The thing I was asking for is no infinitesimals and no calculus, which is a bit tougher I think. But I wouldn't ask if I hadn't seen "partial proofs" which seem to follow that path. – wago Nov 1 '15 at 16:03
• You can't even define "velocity" without using either infinitesimals or proper calculus, I have no idea how you could ever do proper mechanics without them. – ACuriousMind Nov 1 '15 at 16:05
• I agree with Curiousmind – user83548 Nov 1 '15 at 16:07
• I obviously agree with ACuriousMind and Bruce Smitherson. There may be a broader question here, but it would have to start with defining a subset of physics that allows for a formulation of those classical "trajectories" that solve integrable problems (i.e. all half dozen of them) without referencing the dynamics that creates them, i.e. velocity and acceleration are out the window, so are Newton's laws... not sure what's left, then, but such a formulation may still exist. Would it be physics? No. – CuriousOne Nov 1 '15 at 16:10