Classical mechanics in non-commutative phase space

WEI Gao-Feng; LONG Chao-Yun; LONG Zheng-Wen; QIN Shui-Jie; FU Qiang

doi:10.1088/1674-1137/32/5/002

Chinese Physics C> 2008, Vol. 32> Issue(5) : 338-341 DOI: 10.1088/1674-1137/32/5/002

Classical mechanics in non-commutative phase space

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Abstract：
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.

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[1]

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WEI Gao-Feng, LONG Chao-Yun, LONG Zheng-Wen, QIN Shui-Jie and FU Qiang. Classical mechanics in non-commutative phase space[J]. Chinese Physics C, 2008, 32(5): 338-341. doi: 10.1088/1674-1137/32/5/002

WEI Gao-Feng, LONG Chao-Yun, LONG Zheng-Wen, QIN Shui-Jie and FU Qiang. Classical mechanics in non-commutative phase space[J]. Chinese Physics C, 2008, 32(5): 338-341. doi: 10.1088/1674-1137/32/5/002 shu

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Received: 2007-06-29
Revised: 2007-09-26

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通讯作者: 陈斌, bchen63@163.com

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沈阳化工大学材料科学与工程学院沈阳 110142

Classical mechanics in non-commutative phase space

Corresponding author: WEI Gao-Feng,

Received Date: 2007-06-29

Accepted Date: 2007-09-26

Available Online: 2008-05-05

Abstract:

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.

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Classical mechanics in non-commutative phase space

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