What is the tennis racket theorem?
In: 3
I asked the following question here in the comments:
>If we do it in R^n (with n axes of distinct lengths), do we get a stable spin only for the two extreme axes, i.e. shortest and longest?
I need to correct myself
1) Apparently what matters is not the lengths of the axes but values of moments of inertia about the axes
2) I wrongly assumed that in R^n a given axis (a line, 1-dimensional object) alone determines “the” rotation about the axis. This is more complicated for n > 3
First of all let’s talk about *simple* rotations in R^n. These are rotations that have a unique (up to translation) invariant plane called [plane of rotation](https://en.m.wikipedia.org/wiki/Plane_of_rotation). Invariant here means that the image of each point in the plane remains in the plane. In R^n the analogue of the rotation axis from R^3 is the orthogonal complement of the invariant plane, which is an (n-2)-dimensional subspace orthogonal to the invariant plane. In particular **not a line** for n > 3. Let’s call it a generalized axis
A simple rotation doesn’t affect (move) points from its generalized axis. This is analogous to the fact that a rotation in R^3 doesn’t move points on its axis
In R^3 there’s, up to translation, only one plane perpendicular to a given line, so we can identify each rotation with this line, which we call rotation axis. But as I said, it doesn’t work like that in higher dimensions. To talk about a *simple* rotation in R^(n), we need a plane π (then π and its translations will precisely be the invariant planes) and a choice of the “generalized axis” (which is determined by π up to translation; we may choose it to be the orthogonal complement of π or a translation of the orthogonal completent)
Since an invariant plane of a simple rotation is unique only up to translations, it’s convenient to define **the** invariant plane as the unique invariant plane passing through the origin
For example in R^4 **choosing the invariant plane** (any plane, e.g. x=y=0) **and the corresponding generalized axis** , which in the case n=4 will also be a plane (any plane parallel to the orthogonal complement of the invariant plane, i.e. anything parallel to z=t=0), **specifies a simple rotation completely**
Now a general rotation in R^n is a composition of simple rotations such that the corresponding invariant planes are pairwise orthogonal and all meet at exactly one point, the origin*
In particular, given a general rotation in R^(n), it’s possible to choose a basis in which the rotation matrix is block diagonal, with some blocks being 2 by 2 traditional R^2 rotation matrices and the rest just ones
*Note that there are at most floor(n/2) such planes in R^(n). So the lowest dimension in which there exists a non-simple rotation is 4. It’s not so easy to imagine it
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