Classical Mechanics on Noncommutative Space with Lie-algebraic Structure

Details Classical Mechanics on Noncommutative Space with Lie-algebraic Structure Classical Mechanics on Noncommutative Space with Lie-algebraic Structure

2009-11-27

arxiv.org/abs/0911.5227v5

Ann.Phys.326:2091-2107,2011

Physics

Mathematical Physics

21 pages, 4 figures; v2: minor clarification and two references added; v3: 22 pages, clarifications added; v4: 23 pages, clari

Scientific paper

10.1016/j.aop.2011.04.009

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 {\em not only} include that of a recent work as our special cases, {\em but also} provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable $t\dot{x}$-, $\dot{(xx)}$-, and $\ddot{(xx)}$-dependence besides with the usual $t$-, $x$-, and $\dot{x}$-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.

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