Noncommutative version of an arbitrary nondegenerated mechanics

Abstract

A procedure to obtain noncommutative version for any nondegenerated dynamical system is proposed and discussed. The procedure is as follow. Let $S=\int dt L(q^A, ~ \dot q^A)$ is action of some nondegenerated system, and $L_1(q^A, ~ \dot q^A, ~ v_A)$ is the corresponding first order Lagrangian. Then the corresponding noncommutative version is $S_N=\int dt[ L_1(q^A, ~ \dot q^A, \~ v_A)+ \dot v_A\theta^{AB}v_B]$. Namely, the system $S_N$ has the following properties: 1) It has the same number of physical degrees of freedom as the initial system $S$. 2) Equations of motion of the system are the same as for the initial system $S$, modulo the term which is proportional to the parameter $\theta^{AB}$. 3) Configuration space variables have the noncommutative brackets: $\{q^A, ~ q^B\}=-2\theta^{AB}$. It is pointed also that quantization of the system $S_N$ leads to quantum mechanics with ordinary product replaced by the Moyal product.