Quantum subRiemannian dynamics

Abstract

The Hamiltonians that arise in optimal control include some that correspond directly to physical systems and some that do not. For those that do, one can often identify an appropriate quantization procedure and thus associate a Schrodinger equation with an optimal control problem. However, optimal control problems that can not be converted to Lagrangian form, such as those that arise when the system is controllable because of a Lie bracket effect, are more difficult and no straight forward method seems to exist. The purpose of this paper is to describe some possibilities for assigning a Schrodinger equation to such systems. We do not attempt to formulate a general theory but focus on selected examples. We show that this approach leads to interesting "quantum" nonholonomic phenomena. © Springer-Verlag Berlin Heidelberg 2005.