Quantizing Nonholonomic Systems

홍순태, WonTaeKim, 박영재 +1 more

Anthony M. Bloch, Alberto G. Rojo

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.

Oscar E. Fernandez

In this paper we present a quantization procedure for a class of nonholonomic systems—briefly, mechanical systems subject to nonintegrable constraints on the velocities—whose reduced mechanics is Hamiltonian. We illustrate the theory developed through several examples, one of which demonstrates how the classical nonholonomic constraints are sometimes manifested quantum mechanically as a shift in the ground state energy of the system.

Daniel Schubring, Sriram Ganeshan

Nonholonomic mechanics describes systems subject to non-integrable velocity constraints, such as rolling bodies and skating motion. These systems generally lack a canonical Hamiltonian formulation, obstructing standard quantization methods. Here we quantize nonholonomic systems as Markovian open quantum systems, with the nonholonomic constraint appearing in a large-dissipation limit. We find explicit Lindblad superoperators that reproduce the classical dynamics of the Chaplygin sleigh and the Suslov problem in the semiclassical limit. The master equation is numerically simulated, and the covariance is shown to satisfy a relation predicted by the theory of metastability in open quantum systems.

M. S. Marinov

Прохоров Лев Васильевич, Sergei V. Shabanov