I’ve been searching but I could not find any works on the 3D topology of a gyroscope. It seems the spin has a deep origin in the topological and geometrical structures of an elementary particle. Gyroscopic stability is an interesting feature and, in fact, if elementary particles have the topological structure of a gyroscope than it is possible to explain why their spin angular momentum can take half-integral values by applying the Schrodinger wave equation in two dimensional multiply-connected space. Please refer to my articles entitled ON THE SPIN ANGULAR MOMENTUM OF ELEMENTARY PARTICLES, ON THE QUANTISATION OF ANGULAR MOMENTUM and in particular SPACETIME STRUCTURES OF QUANTUM PARTICLES for more details.
Working Paper A TEMPORAL DYNAMICS: A GENERALISED NEWTONIAN AND WAVE MECHANICS
Working Paper SPACETIME STRUCTURES OF QUANTUM PARTICLES
Working Paper A DERIVATION OF THE RICCI FLOW
Working Paper ON THE SPIN ANGULAR MOMENTUM OF ELEMENTARY PARTICLES
http://overunity.com/6455/3-dimensional-gyroscope-shows-antigravity/
https://sundaram.wordpress.com/2013/03/08/mathematical-reason-behind-gimbal-lock-in-euler-angles/
You can describe the instantaneous state of any solid body, like a gyroscope, by its centre-of-mass position r (three cartesian coordinates) and its orientation Ω. The latter can be described by three Euler angles, see the Wikipedia link. Briefly, the latter is equivalent to the surface of a sphere in 4-dimensional space (S3), but with antipodal points identified.
https://en.wikipedia.org/wiki/Euler_angles
Dear Osama and Kare,
Osama: I looked at the references that you recommended but I could not get any information that I need.
Kare: Physically, how can we envisage a 4D sphere with antipodal points identified as a 3D gyroscope? For example, topologically, we can turn cup into a torus. In other words, is it true that a 3D gyroscope does not possess a 3D topology? In this case, if an elementary particle behaves as a 3D gyroscope, it must have a composite structure. It is possible to describe an atom as a system of 3D gyroscopes and an interesting consequence that can be obtained from such description is that that will break the time-reversal symmetry. We are getting older, aren't we? To be honest, I don't like the idea that we are living in a universe that has more than 3 spatial dimensions. That's for pure mathematicians, not for down-to-earth physicists. I might be wrong, but rather be wrong than live in the world of pure mathematics.
Kind regards,
Vu.
Dear Vu
The electron is a free 3D gyro.
And there is a 3D configuration which can yield such behavior.
In the paper On Electric Charge_4_22_2017.pdf duch a topology is discussed.
This topology has been modeled in Matlab and demonstrates spin 1/2 when measured from any direction.
Chip Akins
Vu> how can we envisage a 4D sphere with antipodal points identified as a 3D gyroscope?
Nobody (except you) talked about a 4D sphere (btw., mathematicians usually call it a ball). The surface of such a sphere is 3-dimensional.
How to envisage the procedure of gluing together (identifying) antipodal points? I have difficulty doing that. One can try to consider the same procedure for the 2D-surface of a 3D sphere. Clearly, the southern hemisphere is in some way glued to the northern. So let us forget about it, and concentrate on gluing together antipodal points on the equator. After some trial-and-errors one realises that this process cannot be done in 3D space.
So, I think the best way to envisage the gyroscope is to look a the object itself, and play with it.
The topology is the same as for the rotation group, SO(3). There are some explanations, and presumably a video (which I cannot view on my mac) in the linked page.
http://stephenluttrell.blogspot.no/2007/10/rotation-group-topology.html
Dear Chip and Kare,
Chip: Where to find the paper you mentioned?
Kare: Sorry for the short way to write about a 3D sphere in a 4D space. A single topological structure would make elementary particles more feasible and their dynamics can be formulated entirely in terms of topology. As mentioned in my previous post, I'm afraid that this is not the case and a repeat pattern will continue as described in my work.
Kind regards,
Vu.
Hi Vu
The paper on electric charge can be found here:
https://www.researchgate.net/publication/316260159_On_Electric_Charge
Chip
Dear Chip,
I've read your work on electric charge. It is interesting but hard to see how a topological structure can be fitted into your description of the electron. Could you incorporate my work ON THE SPIN ANGULAR MOMENTUM OF ELEMENTARY PARTICLES into your work so that it would become more acceptable to all followers of the so-called mainstream physics? I am a half-mainstream.
Kind regards,
Vu.
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