Of course, a rotation as a mapping R³ --> R³ does not change distances between points and so also an absolutely undeformable body can rotate. Your question was considered by Helmholtz and Riemann in the 19th century. Helmholtz showed that the free movability of rigid bodies implies that space metric is Riemannian.

@Jan,

are you serious? Our planet Earth an absolutely rigid body?

First manifestation of non-rigidity of Earth rotation is the 40% discrepancy of Euler's prediction (1765) and Chandler's observation (1891) of a periodic motion of Earth's north pole.

@Jan, the question clearly is whether any (even tiny) deformations of the body are required in order to let the body rotate. So 'neglect any inner degrees of freedom' is no option. BTW friction is not the point!

Hello Ulrich, from your comments above it looks like you will object to any object being characterized as being an absolutely rigid body so I will try to avoid that trap. One place that this can be seen is with fundamental particles. I don't know if you have heard about intrinsic spin and spin coupling, but for example an electron which is considered to be a point particle (so there shouldn't be any issues of it being rigid unless you define it away as a case you are not interested in) can have a spin. In fact, in chemistry the study of the intrinsic spin (rotation of the particle) which can occur in either of two orientations spin up or spin down (you can consider this as clockwise or counterclockwise rotation) allows the creation of electron pairs with net spin or a net angular momentum of zero. This of course is the fundamental relationship that leads to all the diversity in the elements and their properties based on filled or partly filled orbitals.

The cleanest way though to talk about objects with a spinning component is to determine the contribution of the rotational kinetic energy which is KEr = 1/2 I w^2 for an object with a moment of inertia I and angular velocity w. As long as a torque can be applied to the object then it will have an angular acceleration and a non-zero angular velocity. There are lots of examples from basic intro physics which demonstrate spinning of rigid bodies. If you look up moment of inertia, that should give you all the background you need to characterize how objects rotate.

Robert,

I don't see how your essay touches the question, and anything I wrote.

I think that the first answer by @Ulrich should have closed the question. This forum need question closing, may be already has it.

@Nadeem

The question refers obviously to classical mechanics. Its last sentence makes the content: as I read it, the author imagines the trajectories of body-fixed points of a rotating ridgid body and sees a surprisingly complicated pattern. He wonders whether these body-fixed points can follow these intriguing lines without being forced to change their relative positions. That they arn't forced to do so is the result of interesting mathematics the historical background of which was addressed in my first contribution. If this helped the questioner to understand that he may still wonder but on the ground of mathematical sassurance, then nothing more had to be said.

Although within the domains of a geometric system idealized rigid bodies are created and may be visualized as undergoing various movement patterns, does this totally translate to the physical Universe? Imagine a 3-dimensional axis system and place a sphere centered at the origin. Observations have shown that a composite body may be induced to rotate. However, every component of the composite body would have a different momentum value for each instant of rotation when considered relative to the reference axis system. Therefore, rotation of a non-rigid sphere represents a blend, tending toward equilibrium, of continually changing momentum components.

Now suppose that an attempt is made to induce rotational motion in a sphere of absolute matter, i.e. a rigid body, by an off-center collision with another identical rigid body. Since collisions should be instantaneously experienced throughout both entire spheres, would it not have the same impact as a dead center, head-on collision; whereas, if composite spheres were utilized, the impulse would migrate from the point of contact inducing translational and rotational impetuses?

William,

what you write makes absolutely no sense to me. To which observations do you refer? Are your centered sphere and your composite body the same? Is there a conclusion or an open problem? What is the purpose of your cryptic story?

Ulrich,

Let me try again.

Step 1. Imagine a static, three-dimensional reference system in a gravity free environment.

Step 2. Place a composite, solid, spherical body (a ball bearing) with its center at the origin of the axis system.

Step 3. Induce the ball bearing to spin so that its center point remains superimposed on the origin. Beyond the center point, every component of the body is continually changing position relative to the established axis of #1.

Step 3. Replace the composite body with a body of absolute matter (Body A) that is not spinning.

Stet 4. In the mind’s eye create an off center collision between Body A and another identical, approaching body of absolute matter (Body B) that is not spinning. Would such a collision result in any rotational motion relative to the axis system of #1?

Step 5. Repeat scenario #4 with ball bearings. Observations have shown the rotational motion may be induced.

In the mathematical/geometrical domain every define body is theoretically rigid; whereas in the observational domain all bodies are composites. In the geometrical domain the points of a rotating sphere maintain their relative positions to each other while continually changing positions relative to a static axis. This system may be utilized to accurately describe rotations of composite bodies such as ball bearings. It does not provide proof that absolutely rigid bodies can rotate. Any body in rotation has different parts of the body traveling in different directions.

Axiom: A body that has no parts is completely rigid. A rigid body cannot undergo distortion, vibrate, rotate, or possess any internal motion. The descriptions of rotations of rigid bodies are paradoxical. Assigning a particle spin assigns it a substructure.

@William,

is abuse of language. There are too many possibly intended meanings to such a statement. It is an unreasonable demand on your reader to confront him with such a, yes, blunder.

@Step 5:

Of course an off center collision of bearing balls (I don't know of spherical ball bearings) will let both balls rotate since during interaction there is friction surface to surface. Reducing friction reduces rotation.

@Step 3: Not only the center, also all points on the rotation axis will not change position. Notice that for a perfect sphere all moments of inertia are equal and thus angular velocity and angular momentum are parallel.

This is right but does not contradict the rigidity of the body as my first contribution tried to convey. Actually, my argument was a bit abbreviated. In full length it is as follows.

Consider a rigid body. Endow it with a body-fixed system of reference formed by a marked point (the origin) and 3 orthogonal axes. Let the body start an arbitrary motion that, however, leaves the origin fixed. Inspect the body at some arbitrarily selected time t. The body-fixed axes are now lie somehow in space and determine three vectors v1(t), v2(t), v3(t), and the original assignment of body-fixed axes may have happened at t=0 so that these axes give vectors v1(0),v2(0),v3(0). Then we form the matrix (R(t))_{ij} = (vi(t).vj(0)), where on the RHS there are scalar products. Since the body is rigid, this matrix R(t) is orthogonal and does not change distances of points. That an orthogonal matrix has this property is a mathematical fact. The reverse conclusion is not plainly correct: also reflections leave the distance between points invariant, but reflections never can arise from motion.

Does this your 'paradox' ? If not, you may start a third attempt to explain it.