What is the tennis racket theorem?
In: 3
Is that also the intermediate axis thing?
If you spin a tennis racket along it’s longest axis (so the handle spins while staying in the same spot), it is happy to spin, and the spin is stable.
If you spin a tennis racket along it’s shortest axis (so it kinda spins flat like a pizza), it is also happy to spin, and is stable).
But if you try to spin it along the intermediate axis (“face” of racket spinning over itself), it is not happy to spin, and is unstable. It will try to adjust itself, but will just end up wobbling in the air, often flipping itself along one of the other axes.
The reason this happens, is because when spun in the intermediate axis, there are internal forces that are only amplified by its spinning.
It’s about how an object that has three different length axis, will behave when it is spun around an axis.
When spun around the longest axis, the spin is stable and it will just spin. When spun around the shortest axis, the spin is also stable and it will just spin as well.
But when the spin is around the third, intermediate axis, then the spin is not stable and you get chaotic behaviour. The math for it isn’t all that simple, but in essence when something is spun around the intermediate axis, any disturbance in movement around the two other axis, gets amplified rather than dampened (as when spinning around the longest or shortest axis) and you get very unstable, chaotic behaviour instead.
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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