Quantization of a Nonholonomic System

In this Letter, we consider the problem of quantizing a nonholonomic system. This is highly nontrivial since such a system, which is subject to nonholonomic constraints, is not variational (or Hamiltonian). Our approach is to couple the system to a field which enforces the constraint in a suitable limit. We consider a simple but representative nonholonomic system, the Chaplygin sleigh. We then quantize the full (Hamiltonian) system. This system exhibits a key complicating feature of some nonholonomic systems—internal dissipative dynamics.

Figure 1

Scheme of the Chaplygin sleigh, a two-dimensional system with a center of mass in $C$ and a pivot at $A$. Point $C$ can rotate freely with respect to $A$, and point $A$ can translate freely along the (instantaneous) direction $CA$ but not along the direction perpendicular to $CA$.