Holonomic constraints and non-holonomic system

[guv]

Homework Statement

A disk of uniform mass density, mass M, and radius R sits at rest on a frictionless floor. The disk is attached to the floor by a frictionless pivot at its center, which keeps the center of the disk in place, but allows the disk to rotate freely. An ant of mass m $\ll$ M is initially standing on the edge of the disk; ou may give your answers to leading order in m/M.

The ant walks an angular displacement $\theta$ along the edge of the disk. Then it walks radially inward by a distance h $\ll$ R, tangentially through an angular displacement −$\theta$, then back to its starting point on the disk. Assume the ant walks with constant speed v.

Through what net angle does the disk rotate throughout this process, to leading order in h/R?

Relevant Equations

The disk will rotate $$\frac{4 m h \theta}{M R}$$

The solution is given. What makes this solution unique is that there is a net turn for the disk. The note of the solution mentions this system is non-holonomic. My question is that are there other non-holonomic examples. What makes this particular set up non-holonomic? Thanks!

 

[haruspex]

Not sure about your holonomics question, but the process is essentially the same as for cat dropped upside down. It spreads out its back legs, pulls in its front legs, and twists. The back legs having greater MoI twist through the smaller angle. It can then swap over the leg postures and twist the other way.
Net result, cat turns in mid air.
Astronauts use the same trick.
For the ant on the disc, it could walk in small circles near one edge of the disc and the disc would gradually rotate the other way.

 

[guv]

That's right, the cat example is also non-holonomic. I think I am starting to get the idea. Looks like I would need to work through the Pfaffian form of the system to rigorously show it's non-holonomic. Any system where the constraint $f(x_i, t) = 0$ is not integrable is non-holonomic because such system's evolution through states depends on the path of evolution.