LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 375z
xFigure 3.2. The roller racer-Tennessee racer.The roller racer is interesting because it generates locomotion. If you climb
aboard and wiggle the joint, the vehicle moves!^1 We will come back to this locomo-
tion question shortly.
The rattleback. This system, also called the wobblestone, is another much
studied example, illustrating the lack of conservation of angular momentum. This
is demonstrated by the reversal of its direction of rotation when spun. General
theory provides a replacement, for the conservation law from "standard" mechanics,
namely the momentum equation.
Figure 3.3. The rattleback.Special features of nonholonomic mechanics. Here are some of the key fea-
tures of nonholonomic systems that set them apart from "ordinary" mechanical
systems:
- symmetry need not lead to conservation laws, but rather lead to an inter-
esting momentum equation, - equilibria can be stable, with some variables being asymptotically stable,
- energy is still conserved,
- Jacobi's identity for Poisson brackets can fail.
To explore the structure of the Lagrange-d'Alembert equations in more detail,
let { wa}, a = 1, ... , k be a set of k independent one forms whose vanishing describes
l(See http://www.isr.umd.edu/ krishna/ for interesting movies of this.)