Chaplygin sleigh

The Chaplygin sleigh is a simple pedagogical example of a nonholonomic system in mechanics, described by Sergey Chaplygin. It consists of a body that slides frictionlessly on a horizontal plane, with a knife edge that constrains its motion so that the knife slides only longitudinally. Because this constraint is nonholonomic, Liouville's theorem does not apply, and although energy is conserved, the motion is dissipative in the sense that phase space volume is not conserved. The motion is attracted to an equilibrium in which the sleigh moves without rotation, with the knife-edge trailing the center of mass.

There are several ways of seeing that the system is nonholonomic. The dimension of the phase space is 5, which is odd. The constraint on the velocity is not derivable from a constraint on the coordinates.

The motion of the system can be characterized simply. Let v be the velocity, with positive values indicating that the knife-edge trails. Let ω be the angular velocity. Let the knife-edge lie at unit distance from the center of mass. Then the equations of motion are of the form

$\dot{v}=\omega^2$

$\dot{\omega}=-b v \omega$ ,

where b is a constant that depends on the mass and moment of inertia. The solutions are ellipses in the v-ω plane. The equations of motion are symmetric under time reversal, but asymmetric under inversion of the body-fixed axis aligned with the knife's edge.

References

A.M. Bloch, Nonholonomic Mechanics and Control