Nonrigid rotation

ImaLooser

I can figure out how to do a rigid rotation in n dimensions. Next I want to look at non-rigid rotations.

Lets says that we have a rotation sphere of fluid in 3D. How mathematically does that rotate?

It is important how viscous the fluid is. If it is infinite viscous them it would essentially be a solid, right? So it seems that there could be a single expression with a viscosity term.

A fluid sphere with zero viscosity would be a superfluid, I think, and hence excluded. Or perhaps not.

ImaLooser

Rotations usually take place in a coordinate system with the center of the sphere at the origin, to avoid unnecessary mess.

Now consider a rotation in four dimensions. Such a rotation has two axis. What if one of the axis does not go through the origin? I think that a sphere would be non-physical and the result would be some sort of ellipsoid instead, but at this point I don't even know what questions to ask. I would guess that a rigid rotation would be impossible, but a fluid rotation would be.

I guess it would depend on the forces that hold the ellipsoid together. I think that gravity alone would not tend to make such a structure, nor would surface tension. It is hard to thnk of what would naturally do this. Maybe a combination of two forces, like gravity and an extremely strong magnetic field (like 10^14 gauss, maybe.) But by this point I'm walking on air like Wile E. Coyote. Anybody have any ideas.

HallsofIvy

What do you mean by "non-rigid rotation"? Any motion (without translation so the center of the object does not move) can be interpreted as a rotation and stretches and so as a "non-rigid rotation".

ImaLooser

Quite so. So there are quite a few situations where it would arise. Let's look at a specific simple example. Let's assume that we have CNIMP (completely non-interacting massive particles). Initially they are evenly distributed through the galaxy. They will be attracted by massive bodies and pass through in cometary or elliptical orbits. It seems that there should be some simple matrix for the case of a single star, though I'm too dumb to figure it out. My best guess is that there is some way to combine position and momentum vectors to get a six-dimensional sphere, then have a rigid rotation of that. Or something like that.

lavinia

I don't know any fluid dynamics but mathematically a vector field that is everywhere orthogonal to the lines from the origin would integrate to a flow that is an instantaneous rotation at every point.