Timeline for Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?
Current License: CC BY-SA 4.0
19 events
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| Dec 15, 2019 at 14:30 | comment | added | Hollis Williams | How does this picture of projecting from $4$-space to ellipses in $3$-space match up with the fact that planets following helical trajectories on a Lorentzian spacetime manifold project down to elliptical orbits in the $3$-space? Presumably discrete symmetry groups don't work the same way in general relativity or have the same meaning? | |
| Nov 9, 2019 at 22:41 | comment | added | 5th decile | Just a random thought: the inverse-square force reminds me of conformal inversion transformations. The extension of $SO(3)$ symmetry to $SO(4)$ and $SO(3,1)$ symmetry reminds me of the way conf$(p,q)$ is isomorphic to $SO(p+1,q+1)$. | |
| Nov 7, 2019 at 8:40 | comment | added | Raphael J.F. Berger | When it comes to the first notion of the SO(4) symmetry in the hydrogen atom one might mention Pauli 1926 as well. | |
| Oct 17, 2019 at 8:03 | comment | added | Tobias Diez | Btw, a nice discussion of this SO(4) symmetry from a symplectic/momentum map point of view can be found in "Global Aspects of Classical Integrable Systems" by Bates and Cushman. | |
| Oct 16, 2019 at 23:24 | comment | added | Terry Tao | Very nice! But something is still bothering me. It seems that this transformation basically transforms inverse square law gravitation (at a fixed energy) to a time-reparameterised version of geodesic flow in the 3-sphere (restricted to a certain invariant subset of the tangent bundle). But I don't recognise the nature of this transformation; it isn't a canonical transformation because of the time reparameterisation. Maybe the question is more naturally phrased in reverse: what high-level fact permits one to transform geodesic flow on $S^3$ to inverse square law gravitation? | |
| Oct 16, 2019 at 22:45 | comment | added | David E Speyer | John Baez's article, linked in the answer above, is great! It seems to me to be the best possible answer. | |
| Oct 16, 2019 at 21:33 | comment | added | lcv | Having seen this, is there now a "low level" explanation? | |
| Oct 16, 2019 at 20:09 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 16, 2019 at 19:56 | comment | added | Michael Engelhardt | I won't presume to edit, but I would know what to do with Z. Phys. 98, 145, whereas I'd be confused by Z. Phys. 98, 3-4. | |
| Oct 16, 2019 at 19:43 | comment | added | LSpice | I think your reference to Z. Phys. 98, 145 was meant to be to Z. Phys. 98, 3–4, assuming that's a reference to the number rather than to the page range (which is 145–154). I edited accordingly, as well as including the name of the paper itself and a link to it. Please feel free to fix it if I messed up the bibliographic information. | |
| Oct 16, 2019 at 19:42 | history | edited | LSpice | CC BY-SA 4.0 |
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| Oct 16, 2019 at 19:14 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 16, 2019 at 19:07 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 16, 2019 at 19:02 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 16, 2019 at 18:54 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 16, 2019 at 18:45 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 16, 2019 at 18:39 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 16, 2019 at 18:07 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 16, 2019 at 18:02 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |