4D rotational symmetry hidden in Newtonian gravity
2nd channel: / @2maniac
4D symmetry in Newtonian gravity? And no, the 4th dimension has nothing to do with time either. Enjoy the finale of this video series on Newtonian gravity. Somehow this SO(4) symmetry is first known in a quantum mechanical setting in the hydrogen atom, before being discussed in the classical setting. For hyperbolic orbits, it will be a SO(3,1) symmetry, which suggests some relations to special relativity, but there isn't a very direct relation.
This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.
Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: SO4gravity
Further reading:
1) John Baez's write-up on this, and expansion further:
https://arxiv.org/pdf/2507.11428
or start from his blog posts here:
https://johncarlosbaez.wordpress.com/...
2) Circular hodograph geometric explanation:
http://butikov.faculty.ifmo.ru/Hodogr...
3) Most materials are just from Wikipedia:
https://en.wikipedia.org/wiki/Laplace...
4) 4D hyperbolic "rotation" - this paper is also doing quantum mechanics: https://iopscience.iop.org/article/10...
5) Credit to the photo of Fock I used: https://en.wikipedia.org/wiki/Vladimi...
Video chapters:
00:00 Introduction
00:38 Chapter 1: Why velocity / momentum trace out a circle
06:47 Chapter 2: What is symmetry?
10:35 Chapter 3: Circular hodograph (Part 2)
16:02 Chapter 4: 4D, finally
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