A disk rolling on a horizontal surface without slip

A circular disk rolling on a horizontal surface without slip is a common example of a nonholonomic problem of Lagrangian mechanics. In the literature, very few such problems have been attacked developing simple and interesting results. Here, it will be shown first that it is possible to find a steady motion of the system, which is rolling in a circle. Then it will be shown that it is possible to find the expression for the frequency of small oscillations about this steady motion, including an approximate expression for the speed required for stability of that steady motion, which is accurate for circles large in radius compared to the disk. The solution is shown to reduce to the well-known critical speed for a disk rolling in a straight line. It is then surprising to note that the critical speed for stability is slower for motion in a circle than for the straight line. This result will be developed entirely analytically which seems to be quite unusual for such a nonholonomic problem of this complexity.