Newton's theorem of revolving orbits | Wikipedia audio article
Channel: wikipedia tts
Date: 27/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Newton%27s_theorem_of_revolving_orbits


00:01:34 1 Historical context
00:08:14 2 Mathematical statement
00:10:27 2.1 Alteration of the particle path
00:11:48 3 Orbital precession
00:12:55 4 Illustrative example: Cotes's spirals
00:13:45 5 Closed orbits and inverse-cube central forces
00:14:59 6 Limit of nearly circular orbits
00:16:05 6.1 Quantitative formula
00:20:41 6.2 Examples
00:20:58 7 Precession of the Moon's orbit
00:22:35 8 Generalization
00:22:48 9 Derivations
00:23:05 9.1 Newton's derivation
00:23:32 9.2 Modern derivation
00:25:57 10 Newton’s Geometric Proof from the Principia
00:26:09 10.1 Simplified Geometric Proof of Proposition 44
00:28:05 10.2 Newton’s Proof of Proposition 44
00:28:08 10.3 Newton’s Proof of Proposition 45
00:28:18 10.4 Proposition 6 for Proof of Proposition 44, above
00:29:18 11 See also
00:29:30 12 References
00:29:48 13 Bibliography
00:30:52 14 Further reading
00:34:56 15 External links
00:35:40 k θ1. In contrast to Newton, however, Mahomed and Vawda did not require that the radial motion of the two particles be the same, r1
00:38:48 1 and b
00:38:59 r2. In this case, the original force is not scaled, and its argument is unchanged; the inverse-cube force is added, but the inverse-square term is not. Also, the path of the second particle is r2
00:39:27 Derivations
00:39:37 Newton's derivation
00:47:01 Modern derivation
00:54:09 Newton’s Geometric Proof from the Principia
00:54:21 Simplified Geometric Proof of Proposition 44
01:07:58 Newton’s Proof of Proposition 44
01:14:32 Newton’s Proof of Proposition 45
01:22:40 Proposition 6 for Proof of Proposition 44, above
01:27:26 See also



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SUMMARY
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In classical mechanics, Newton's theorem of revolving orbits identifies the type of central force needed to multiply the angular speed of a particle by a factor k without affecting its radial motion (Figures 1 and 2). Newton applied his theorem to understanding the overall rotation of orbits (apsidal precession, Figure 3) that is observed for the Moon and planets. The term "radial motion" signifies the motion towards or away from the center of force, whereas the angular motion is perpendicular to the radial motion.
Isaac Newton derived this theorem in Propositions 43–45 of Book I of his Philosophiæ Naturalis Principia Mathematica, first published in 1687. In Proposition 43, he showed that the added force must be a central force, one whose magnitude depends only upon the distance r between the particle and a point fixed in space (the center). In Proposition 44, he derived a formula for the force, showing that it was an inverse-cube force, one that varies as the inverse cube of r. In Proposition 45 Newton extended his theorem to arbitrary central forces by assuming that the particle moved in nearly circular orbit.
As noted by astrophysicist Subrahmanyan Chandrasekhar in his 1995 commentary on Newton's Principia, this theorem remained largely unknown and undeveloped for over three centuries. Since 1997, the theorem has been studied by Donald Lynden-Bell and collaborators. Its first exact extension came in 2000 with the work of Mahomed and Vawda.