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In the Wikipedia article for "Kepler's laws of planetary motion," the article suggests that "Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation."

A commentator on another question suggests that the "kepler's laws' article is not correct on this. Newton's Principia of 1687 showed (Bk1 sec.11 props 57-69, esp 66) that in a system with >2 masses, deviations from keplerian motion would occur, though they might be rather small if one of the bodies was of much greater mass than the others"

Please could someone describe the relationship of the Newtonian and Keplerian laws that Newton was able to achieve (if it's different from the above description by the previous commentator) and maybe also give a brief description of how the laws were used together (if they were) after Newton? Please let me express my gratitude for your consideration in advance!

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    $\begingroup$ I suggest that the way you've formed the question almost rules out the possibility of any answer, because the cited sections in 'Principia' do show what you quoted. But nobody suggests that Newton 'combined' any laws with Kepler's 'laws'. The position is rather that Kepler's 'laws' are approximations to the motions, they are not exact. $\endgroup$
    – terry-s
    Commented Dec 23, 2023 at 13:24
  • $\begingroup$ @terry-s, thanks for the comment! I've emended the question according to your concern about the words "combination" and "unified." Does the question now have the possibility of an answer, please? $\endgroup$
    – Colin Pace
    Commented Dec 23, 2023 at 15:48

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Newton's law of gravity is equivalent to Kepler's laws in the following sense: for two bodies, assuming attraction by a central force if the force obeys Newton's law then the motion obeys Kepler's laws; and conversely, Kepler's laws imply Newton's law of attraction. This is a mathematical theorem essentially established by Newton.

For more than 2 bodies, there are no "Kepler laws"; for example, for the system Sun-Earth-Moon, Kepler's law hold only approximately, if we neglect the influence of one of the bodies on the two others. The motion under the Newton's law of attraction is in general very complicated. And it took long time to derive all observed motions in the Solar system from Newton's gravity laws.

Consistence of the theory with observations is excellent, except for Mercury, where a more general gravity law is required - general relativity.

Remark. There was a long and heated discussion in 20th century, whether Newton really proved equivalence of his gravity law to Kepler laws, or only an implication in one direction. On my opinion, he proved the equivalence.

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  • $\begingroup$ Thank you for the clear answer about the relationship between Newtonian and Keplerian laws, Alexandre! Please, could you cite the area of "Principia" where the "mathematical theorem essentially established by Newton" is? $\endgroup$
    – Colin Pace
    Commented Dec 23, 2023 at 17:08
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    $\begingroup$ Book I, sections 2-3. Propositions 1, 11, 15 are the second, first and third Kepler's laws. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 24, 2023 at 15:41
  • $\begingroup$ @Alexandre Eremenko : (on the question where, in Principia, are the deductions of Kepler's laws for a 2-body system) It's more complicated than that. Book 1 lays mathematical foundations, but doesn't assert whether the indicated conditions occur in the real world. And in Newton's habitual piecemeal way of assembling results, not only prop 15 for the third 'law', but also prop 4 corollary 6 contributes too. It is only later, in Book 3, that Newton discusses the observations that support conclusion that conditions really exist for applicability of those theoretical results of Book 1. $\endgroup$
    – terry-s
    Commented Dec 25, 2023 at 1:16

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