Cat righting reflex: Is the cat's angular speed zero or non-zero? (Or is it more complicated?)

Question

The cat righting reflex (Wikipedia article) allows a cat to change its orientation in the absence of initial angular momentum or external forces. A theoretical model would work like this: the cat bends its spine $90^{\circ}$, then performs the maneuver in the animation below (also from Wikipedia) ...

... then straightens its spine, thereby arriving at a position that is an exact $180^{\circ}$ rotation of its initial orientation.

Now, assume the cat begins its fall with no initial angular momentum. We know that angular velocity is a linear function of angular momentum, which would suggest angular velocity is zero. The cat also changes its orientation, and we know that angular velocity is linearly related to the derivative of orientation (depending on how we represent orientation, and whether we look at it in 2D or 3D), and this would suggest angular velocity is not zero. We could define orientation as the vector between the cat's middle of the tummy its middle of the back (at the crease of the bend), so that this remains a well-defined quantity throughout the spine bend and the rotation.

So is angular velocity zero or not? Or are there multiple ways in which we could define it, which could provide different understandings of the motion?

Answer 1

The cat is not a solid and therefore we cannot define a single angular velocity: each point has a different angular velocity. And it can be nonzero even if the total angular momentum is zero.

Answer 2

Cats are not rigid bodies, and hence defining the angle of rotation from the starting configuration is somewhat arbitrary. You can't rotate back by some unambiguous angle until the shape superimposes the starting configuration, because the shape has likely changed, leaving you some wiggle room for many options that all seem appropriate to define the angle of rotation.

This ambiguity has to be removed by defining what you mean by rotation. This same ambiguity is shared by translation. It is common to define translation by the center of mass, but it's far from the only option (you could use the geometric centroid without mass weighting for example). Similarly, rotation is often removed by using a geometric least-squares fit between sites, a mass-weighted fit, or by going to the principal axes frame, etc, all of which give you slight different angles and slightly different angular velocities. Some of these frames will have desirable properties (much like the center of mass frame for translation) but they are ultimately arbitrary.

Now, one consequence of all this is that most of these frames do not actually eliminate the angular momentum in the co-rotating frame. So it becomes possible to define rotational angular momentum and vibrational angular momentum separately, only the former of which is connected to the angular velocity. One way to think about the falling cat problem is that the cat is allowed to generate some rotational angular momentum, and hence a nonzero angular velocity, provided that it cancels it out with equal and opposite vibrational angular momentum. Hence the twisty acrobatics.

The interpretation for how the cat turns around can appear slightly different depending on how you define the rotation of the cat in the first place, which is why this topic can get somewhat confusing. But the physics is the same in all cases. The net angular momentum remains zero, but angular momentum isn't the same thing as angular velocity. Nonrigid bodies can take advantage of coriolis, euler, and centrifugal forces that couple vibrational degrees of freedom with rotational degrees of freedom to exploit this and generate angular velocity without angular momentum.