Classical mechanics in non-commutative phase space

• Published in: Chinese physics C 2008-05, Vol.32 (5), p.338-341
• Main Author: 卫高峰 龙超云 隆正文 秦水介 付强
• Format: Article
• Language: English
• Subjects: 不可交换相空间; 动量交换; 古典力学; 微粒物
• ISSN: 1674-1137; 0254-3052; 2058-6132
• DOI: 10.1088/1674-1137/32/5/002

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space. The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity. First, new Poisson brackets have been defined in non-commutative phase space. They contain corrections due to the non-commutativity of coordinates and momenta. On the basis of this new Poisson brackets, a new modified second law of Newton has been obtained. For two cases, the free particle and the harmonic oscillator, the equations of motion are derived on basis of the modified second law of Newton and the linear transformation （Phys. Rev. D, 2005, 72： 025010）. The consistency between both methods is demonstrated. It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space, but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.