Re: Rotation in N-space

On Sun, 19 Jun 2005, Neil wrote:

<tessel@xxxxxxx> wrote

On Sat, 14 May 2005 frisbieinstein@xxxxxxxxx asked:

Consider a rigid N-solid in N-space.  What are the possible dimensions
of the axis of rotation, and is it possible to have two simultaneous


You may ultimately be asking about a higher dimensional analog of the
Euler equations describing a tumbling rigid body in E^3, but right now it
seems you seek the "orthogonal decomposition" of an element of SO(n,R)
into a -commuting- product of "ordinary rotations".

Try something simpler and more fun:

I think the decomposition I mentioned is simple, useful, and fun. If I made it seem otherwise, that reflects my failings as an expositor.

Truth to tell, Neil, I myself did not find -your- post very easy to understand. But FWIW, when you say that in E^4,

we can have two simultaneous rotational planes,

you are talking about the orthogonal direct sum decomposition of R^4 (as a euclidean inner product space), wrt some rotation or uniparametric subgroup of rotations, into two two-dimensional invariant subspaces A,B, and the fact that, wrt a suitably "adapted basis", a generic rotation in E^4 has a matrix of form

	[ R 0 ]
        [ 0 S ]

where R,S are either both in SO(2) or both in O(2).

(You are talking about the decomposition of unidimensional subgroups of SO(4), whereas I was talking about -elements- of SO(4), but this makes very little difference.)

When you say

That means that on a 4-D planet, there can be two periods of rotation

you are saying that wrt an adapted basis, the elements of a generic unidimensional subgroup in SO(4), parameterized by t, have the form

        [ 0 -a 0  0 ]   [ cos(at) -sin(at)   0        0     ]
  exp t [ a  0 0  0 ] = [ sin(at)  cos(at)   0        0     ]
        [ 0  0 0 -b ]   [    0        0    cos(bt) -sin(bt) ]         (*)
        [ 0  0 b  0 ]   [    0        0    sin(bt)  cos(bt) ]

or else a very similar form corresponding to a pair of generators in O(2) rather than SO(2). Here, a,b are the two frequencies. The orbits of the "ordinary rotation" corresponding to the upper left block are concentric circles in the 2-flats orthogonal to the three dimensional subspace

	x_3 = x_4 = 0

That is, the orbits of the first ordinary rotation in the decomposition are a three parameter family of circles which you can work out from

  [ cos(at) -sin(at)  0  0  ] [ x_1 ]   [ x_1 cos(at) - x_2 sin(at) ]
  [ sin(at)  cos(at)  0  0  ] [ x_2 ]   [ x_1 sin(at) + x_2 cos(at) ]
  [   0        0      1  0  ] [ x_3 ] = [            x_3            ]
  [   0        0      0  1  ] [ x_4 ]   [            x_4            ]

(The obvious parameters of the family of orbits are of course x_3, x_4, and r = sqrt(x_1^2 + x_2^2). Thus, "concentric circles in the family of two flats orthogonal to the other invariant subspace", or in still higher dimensions, to the direct sum of all the other invariant subspaces.) Similarly for the lower right block.

But of course the orbits of the -composition- are much more interesting, and IIRC, in my post I originally intended to set this

Exercise: describe the orbits of the unidimensional subgroup (*). Discuss.

[Hint: the orbits form another three parameter family of curves in E^4. How does your answer depend upon the two frequencies a,b?]

(Spoiler below!)

(I must have removed the material on unidimensional subgroups before submitting my post because I've said all this here so many times before that I guess I've grown bored with explaining it. Sorry about that!)

"Planet": Newtonian gravitation in higher dimensions would be quite different from E^3, as Newton knew, and our discussion here is not really terribly relevant to that problem. See the problem on "force laws" in higher dimensions in Landau & Lifschitz, Mechanics.

different durations.

Such an action would not longer correspond to a uniparametric subgroup, but yes, there is an obvious way to turn R actions into R^m actions in cases like this, and then you can choose to advance one parameter while keeping another fixed, then advance both parameters, etc. (Here, m is the number of independent periods [or frequencies] in the original unidimensional subgroup.)

These unidimensional subgroups are geodesics passing through the identity element of our group SO(n). (This claim of course refers to an appropriate metric on the manifold underlying the Lie group.) Of course, one can consider other curves through the identity element, but these will almost never have a fixed configuration of invariant subspaces. This is pretty much the same thing as saying that if we compose two generic elements of SO(n), we'll get a third generic element of SO(n), but the three decompositions, while having similar dimensional characteristics, will almost never be even partially aligned.

So, we could have a day of 24h and 33h combined on one planet. In a 5-D world we can have two rotations mutually around a line, 
                    ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Not sure I follow that. "Circles around a line" doesn't make dimensional sense in E^n when n > 3 (see the last chapter of Colin Adams, The Knot Book for an illustrated discussion of topological circles up to diffeotopies of R^4 and R^5.). Compare with the phrase I used, "concentric two-circles in two-flats parallel to a two-dimensional invariant subspace", which does make dimensional sense.

You probably mean that a generic unidimensional subgroup of SO(5) has an orthogonal decomposition into two unidimensional subgroups of ordinary rotations. Such a group has one fixed line, two two-dimensional invariant subspaces, and two frequencies (a,b), or equivalently two periods. We can then lift from the given R action to an R^2 action via (at,bt) -> (u,v). We now have a direct product of our two subgroups of ordinary rotations. This gives a two dimensional abelian subgroup of SO(5), hence my phrase "orthogonal decomposition into commuting rotations".

in 6-D we can have 3 independent rotations, etc. 
                       ^^^^^^^^^^
                        commuting ordinary

A general unidimensional subgroup of SO(6) has no fixed line, three two-dimensional invariant subspaces, and three frequencies a,b,c. We can then lift from the given R action to an R^3 action via (at,bt,ct) -> (u,v,w). This gives the direct product of our three unidimensional subgroups of ordinary rotations, which is a three dimensional abelian subgroup of SO(6).

More generally, we get an m-dimensional abelian subgroup of SO(n), where m is the maximal number of independent periods. The nature of a unidimensonal subgroup of this abelian subgroup depends upon--- well, I'll give a hint about that at the end.

It's rather weird but I don't see much about it in recreational math books.

Well, they mostly shy away from anything involving matrices or other "advanced" math ;-/

This material can be massaged into statements about harmonic oscillators, generalized Penrose tilings, Clifford algebras, and various other things of independent interest. If you look through several decades of back issues of the American Mathematical Monthly, you will find at least one expository paper mentioning these ideas.

(I could give the citation, but it's probably more pedagogically sound to make anyone interested look through several decades of back issues of the Monthly! You should find several other very cool papers related to rotation--- be sure to read the article by Altman on quaternions and the one by Howe on Lie theory.)

"T. Essel"

HINT : as we have discussed in the past in this n.g., the topology of the orbits depends upon a tiny bit of number theory! Namely,

* If a,b are rationally related, the orbits are topological -circles-, each lying in a certain torus in the unit three sphere S^3. In fact, the orbits become "torus knots" when you stereographically project from S^3 to R^3. But in R^4 the orbits are unknotted, for dimensional reasons; see the book by Adams.

* If not, the orbits are topological -lines- lying in a torus on S^3. But note well: the orbits are curves which are -dense- in each torus. In fact, this case corresponds to a "quasiperiodic" unidimensional subgroup of rotations in E^4, and you may see here a hint of the promised connection to quasiperiodic tilings.

These tori are in fact the "Hopf tori" appearing in the famous Hopf fibration of S^3 by circles. They form a configuration of tori nested about two great circles which -are- knotted on S^3. Each Hopf torus in locally -flat- (vanishing Gaussian curvature) as a submanifold of S^3.

This can all be generalized in various directions.

.

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