newtonian mechanics - Is there a way for an astronaut to rotate?

We know that if an imaginary astronaut is in the intergalactic (no external forces) and has an initial velocity zero, then he has is no way to change the position of his center of mass. The law of momentum conservation says: $$ 0=\overrightarrow{F}_{ext}=\frac{d\overrightarrow{p}}{dt}=m\frac{d\overrightarrow{v}_{c.m.}}{dt}$$

But I don't see an immediate proof, that the astronaut can't change his orientation in the space. The proof is immediate for a rigid body (from the law of conservation of angular momentum). But the astronaut is not a rigid body.

The question is: can the astronaut after a certain sequence of motions come back to the initial position but be oriented differently (change "his angle")? If yes, then how?

Answer

The astronaut can change his or her orientation in the same way that a cat does so whilst falling through the air. After the transformation, the astronaut is still and angular momentum is conserved. There is a rather beautiful way of understanding this rotation as an anholonomy i.e. a nontrivial transformation wrought by the parallel transport of the cat's (or astronaut's) state around a closed loop in cat configuration space. I'll write a bit more about this when I have some more time, but for now, one can give a simple explanation with an idealized "robot cat" (or astronaut) which I made up for the thought experiment:

Above I have drawn a simplified cat. I am a very aural person, so this is good enough for me so long as I can imagine it mewing!

Now our "cat" comprises two cylindrical sections: the "forecat" (F), "hinder-cat" (H) and two legs (L) which can be drawn in so that they are flush with the hinder-cat's surface. With the legs drawn in, the forecat on one hand and hinder-cat + legs assembly on the other have the same mass moment of inertia about the axis of the body. Here is how the cat rotates:

1. Deploy legs symmetrically, i.e. spread them out as shown in the drawing. Now the hinder cat + legs has a bigger mass moment of inertia than the fore cat. Note that, if the legs are diametrically opposite and identical and are opened out symmetrically, the cat undergoes no motion;


2. With an internal motor, the forecat and hinder cat exert equal and opposite torques on one another to accelerate, then stop. Owing to the differences between the moments of inertia, the forecat undergoes a bigger angular displacement than the hind cat;


3. Pull the legs. Again this begets no motion if done symmetrically;


4. Use the internal motor again with an acceleration / deceleration sequence to bring the forecat and hinder cat back to their beginning alignment (i.e. with the line along the cylinders aligned). Now the two halves have the same mass moment of inertia, so when the cat is aligned again, the rotation angles are equal and opposite.

Since the rotation angles are different in step 2, but the same in step 5, our robot cat's angular orientation has shifted.

If you want to know more about the "Berry phase" explanation and the anholonomy of the cat configuration space before I get around to expanding on this, see Mathematics of the Berry Phase by Peadar Coyle. This is not peer reviewed, but looks sound and is in keeping with similar treatments along these lines that I have seen.