Classical mechanics on noncommutative space with Lie-algebraic structure

Journal Article · Mon Aug 15 00:00:00 EDT 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2011.04.009· OSTI ID:21583329

Miao Yangang, E-mail: [e-mail] ^[1]; Department of Physics, University of Kaiserslautern, P.O. Box 3049, D-67653 Kaiserslautern ^[2]; Bethe Center for Theoretical Physics and Institute of Physics, University of Bonn, Nussallee 12, D-53115 Bonn ^[2]; Wang Xudong, E-mail: [e-mail] ^[1]; Yu Shaojie, E-mail: [e-mail] ^[1]

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1. School of Physics, Nankai University, Tianjin 300071 (China)
2. (Germany)

Highlights: > Suggest a useful method to look for new Lie-algebraic noncommutative spaces. > Find out two new Lie-algebraic noncommutative spaces. > Derive Newton and Hamilton equations that present unimaginable extra forces. > Analyse the source of unimaginable extra forces from space noncummutativity. > Provide various intriguing classical trajectories. - Abstract: We investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results not only include that of a recent work as our special cases, but also provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable tx-dot-,(xx-dot)-, and (xx-double dot)-dependence besides with the usual t-, x-, and x-dot-dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle's ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.

• OSTI ID: 21583329
• Journal Information: Annals of Physics (New York), Vol. 326, Issue 8; Other Information: DOI: 10.1016/j.aop.2011.04.009; PII: S0003-4916(11)00068-6; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CLASSICAL MECHANICS
COMMUTATION RELATIONS
EUCLIDEAN SPACE
HAMILTONIANS
KINETICS
LIE GROUPS
TRAJECTORIES
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
QUANTUM OPERATORS
RIEMANN SPACE
SPACE
SYMMETRY GROUPS