4d gyroscope

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4d gyroscope

Postby granpa » Wed Feb 25, 2015 9:53 pm

How to calculate precession for a 4d gyroscope?
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Re: 4d gyroscope

Postby granpa » Fri Apr 10, 2015 10:30 pm

3000 views on only one post! there seems to be some interest in this topic.

I've been giving it some more thought.

in 4 spatial dimensions an object can rotate in two different ways simultaneously.
setting one rotation to 0 and just looking at one rotation alone and assuming that there is a preferred plane of rotation, that is to say that any tilt away from that plane results in a restoring force back toward that plane then it seems pretty clear that a simple tilt will result in the normal gyroscopic precession that we are all familiar with but in 4 spatial dimensions the plane of rotation can tilt in two different ways simultaneously. It seems probable therefore that you would get two different processions but they would both be at the same frequency so if an atom were precessing in this way you probably wouldn't even notice it.

when you make the second rotation nonzero my first thought was that you would get a very complex motion involving four different precessions.
But the two rotations are not independent.
Any precession of one rotation is necessarily a precession of the other rotation. So it seems to me that the 2 angular momentums will simply add.
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Re: 4d gyroscope

Postby granpa » Fri Apr 10, 2015 11:01 pm

We know the magnetic moment and angular momentum that an atom has but when we measure it in a strong magnetic field we find that that the z component is less than that. It is as though the atom were coming to rest aligned not parallel to the magnetic field but rather at an angle to the magnetic field. This makes no sense in three dimensions where it should always come to rest aligned perfectly parallel to the magnetic field. But it does make sense in four spatial dimensions where the atom rotates in two different ways simultaneously.
Since there are two rotations it is obviously impossible for both rotations to come to rest aligned parallel to the magnetic field.
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Re: 4d gyroscope

Postby ICN5D » Sat Apr 11, 2015 2:55 am

It certainly is an interesting idea. If I understand things correctly, it seems like if the gyroscope was in isoclinic double rotation, there would be no precession at all, since it has reached maximum equilibrium. Any deviation in rotation speed is what would induce the precession. So, no matter how much wobble it currently has, it eventually tapers out. It'd be interesting to figure out, but I'm not familiar with the maths.
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Re: 4d gyroscope

Postby wendy » Sat Apr 11, 2015 6:48 am

The principle of equipartition of energy holds, and energy would be transferred.

A freely rotating sphere would resolve into either left or right handed clifford rotation, which pretty much renders it useless s a gyroscope. You see, there are no distinct axies of rotation in a clifford rotation. Things would probabaly be different for xOoOxOo, a bicircular ellipsoid.

I'm still not sure about something rotating at oblique angles. John Conway tells me that a rotation is the product of left and right hamiltons, and because hamiltons are governed by a trimex, they are A(BC)=(AB)C, so the only way out of this is for the separate rotations to be orthogonal.
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Re: 4d gyroscope

Postby granpa » Sat Apr 11, 2015 12:14 pm

well obviously the two rotations are orthogonal.

Since conservation of angular momentum holds for 4 dimensions it follows that energy will not be transferred between the two rotations.
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Re: 4d gyroscope

Postby granpa » Tue Nov 17, 2015 10:03 am

There is no axis of rotation in four dimensions but there is a plane of rotation
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Re: 4d gyroscope

Postby PatrickPowers » Thu Dec 10, 2015 3:55 pm

granpa wrote:How to calculate precession for a 4d gyroscope?


Here's what I think. A 4D gyroscope would have an axle in the shape of an X. (I call it an axle with two subaxles.) The axle would touch the "ground" in two spots. This constrains the rotation to a plane. The rotor still has one degree of freedom to precess.

It is easy to visualize. Just imagine one of the subaxles going off into a dimension that you can't see. Then it all reduces to the familiar case.

This should scale up to any number of dimensions. There are N-2 subaxles to an ND axle.
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