Newton equation for canonical, Lie-algebraic, and quadratic deformation of classical space

Abstract

The Newton equation describing the particle motion in constant external field force on canonical, Lie-algebraic and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of particle is generated. We also indicate, that in the case of spatial coordinates commuting in Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and positiondependent forces.1 There also exist so-called fuzzy space noncommutativity [43]. However, in this article such a type of deformation will be not under consideration.2 We consider only spatial deformations, i.e. time plays a role of parameter. 3 In the case of "position force" one can recognize well-known inverted oscillator force (see e.g. [46]).

Scalar field theory on noncommutative Snyder spacetime

“…This transformation can be a generic function of momenta, but linear in coordinates (for discussions on noncommutative classical mechanics see e.g. [44]). …”

Classical mechanics on noncommutative space with Lie-algebraic structure

“…There has been a large amount of literature on noncommutative quantum mechanics (NCQM) [3] and noncommutative field theory (NCFT) [4] as well. However, we notice that the research on noncommutative classical mechanics 1 (NCCM) [5,6,7], if comparing with that of NCQM and NCFT, is so little. Probably the NCCM is not so attractive as the NCQM and NCFT; nevertheless, the Doubly Special Relativity [7] has been intriguing.…”

Para-Galilean Versus Galilean Noncommutative Phase Spaces

“…In both cases, the angle α is constant in time. Moreover, in the case of canonically deformed phase space it has been proved in [9] that for Hamiltonian function of the form…”