Chaplygin sleigh

Example

The Chaplygin sleigh is a rigid body that moves on a horizontal plane. It rests on the plane at a knife edge and at other points that slide without friction. The knife edge forbids motion of its contact point perpendicular to the blade. Let $(x, y)$ be the contact point of the blade and $\theta$ the angle of the blade with the $x$-axis. Let $m > 0$ be the mass and $I > 0$ the moment of inertia about the centre of mass. Let $a \ne 0$ be the signed distance from the contact point to the centre of mass, measured along the blade. The configuration space is $\R^2 \times S^1$ with coordinates $(x, y, \theta)$, and the constraint is $$-\dot x \sin\theta + \dot y \cos\theta = 0.$$ The vector fields $(\cos\theta, \sin\theta, 0)$ and $(0, 0, 1)$ span the allowed velocities, and their Lie bracket $(-\sin\theta, \cos\theta, 0)$ is not in their span. So the constraint is not integrable, and the sleigh is a nonholonomic system.