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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?

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wago
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    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 01 '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 01 '15 at 16:03
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    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 01 '15 at 16:05
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    I agree with Curiousmind –  Nov 01 '15 at 16:07
  • That's why I was suggesting to take them as primitive, to leave the questions about them behind. Beware - I'm not talking about rephrasing classical mechanics without infinitesimals and calculus, that's most probably impossible: I'm just talking about a path from inverse-square law to Kepler's laws. – wago Nov 01 '15 at 16:08
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    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 01 '15 at 16:10
  • I think you are missing the point. Take a look at the article I mentioned: it derives conservation of angular momentum (which couldn't probably be definied from scratch without calculus) for a central force without using any calculus or infinitesimals. Is that non-physics? Non-dynamics? Of course not. It's a deriviation of a specific property which doesn't use infinitesimals or calculus in any of its deductions. Just like in many mathematical systems with axioms A, B, and C you can happen to derive proposition P from proposition Q using just A and B, even if you used C to derive P before. – wago Nov 01 '15 at 16:16
  • That's what I just said: conserved variables of that kind only exist for integrable systems and there are very few of those. If you restrict yourself to a geometric theory of integrable systems, only, Hamiltonian mechanics leaves you stranded with a handful of next to trivial examples. OTOH, maybe that's exactly what you are asking for? In that case there is a pretty good theory for you out there, already. I would start by reading the papers and textbooks of the mathematicians who are working on symmetry, integrability and geometry. There are journals and conferences out there, too. – CuriousOne Nov 01 '15 at 16:20
  • @wago Having said that, it is also true that you can skip calculus. Any know mathematical theory (with the exception perhaps of homotopy type theory) can be interpreted within set theory. That is, all you have are sets, not even numbers (they can be defined from sets). But it will not look like anything you have in mind. –  Nov 01 '15 at 16:23
  • Newton's own proof in Principia uses neither modern calculus nor infinitesimals, only kinematic notion of "first and last ratios", which sounds like "assuming concepts such as velocity and acceleration as primitive". One can also rework any geometric limit proof into a double reductio a la "method of exhaustion", would that work for you? – Conifold Nov 01 '15 at 20:48
  • Hm, in prop. 1, theorem 1 (his Proof of Kepler's Second Law) Newton writes: "Now let the number of those triangles be augmented, and their breadth diminished in infinitum"... This seems direct use of infinitesimals to me. About exhaustion... Geometrically speaking it's perfectly fair, but - if you have to - how do you compute the final sum of the ever-decreasing parts without using infinitesimals? Some times you can - Archimides's exhaustion proof for the area of a parabolic segment does that in a purely geometrical way - but I'm not sure it's always possible (or is it obvious that it is?). – wago Nov 01 '15 at 21:00
  • Newton defines magnitudes in kinematic terms, so he is allowed to say "their breadth diminished in infinitum" without infinitesimals or Weierstrassian limits. This is subtle, see p. 478-481 http://izt.ciens.ucv.ve/ecologia/Archivos/Filosofia-II/Friedman,%20Michael%20-%20Kant's%20theory%20of%20geometry.pdf For exhaustion you assume that the final sum is less than suspected value by something, then find a partial sum which is greater than that, then the same assuming it is greater than suspected value. It should work as long as the sum is "constructive". – Conifold Nov 01 '15 at 21:26
  • Yes, you're definitely right. Thanks for the link. The main argument is something I read about before but I somehow forgot to take into account when formulating my discourse. Anyway, the fact that Newton wanted or not to use infinitesimals/vanishing quantities doesn't alter the fact that many of his quasi-infinitesimal proofs don't meet today's rigour standards. The main point of my questions is probably just this one: in order to derive his results rigorously (in some viable modern sense of rigour), is calculus the only possible strategy? Or are there other methods - such as indivisibles? – wago Nov 01 '15 at 21:56
  • I am not familiar with indivisibles, but there are intuitionistic versions of calculus and differential geometry that are rigorous and closer to Newton's kinematic conception than to Weierstrass's. http://www.fuw.edu.pl/~kostecki/sdg.pdf For that matter, there is even a constructivist version of infinitesimals. https://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/ – Conifold Nov 02 '15 at 00:41

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