Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 028, 9 pages      hep-th/0610061      https://doi.org/10.3842/SIGMA.2008.028
Contribution to the Proceedings of the 3-rd Microconference Analytic and Algebraic Methods III

Noncommutative Lagrange Mechanics

Denis Kochan a, b
a) Dept. of Theoretical Physics, FMFI UK, Mlynská dolina F2, 842 48 Bratislava, Slovakia
b) Dept. of Theoretical Physics, Nuclear Physics Institute AS CR, 250 68 Rez, Czech Republic

Received November 26, 2007, in final form January 29, 2008; Published online February 25, 2008

Abstract
It is proposed how to impose a general type of ''noncommutativity'' within classical mechanics from first principles. Formulation is performed in completely alternative way, i.e. without any resort to fuzzy and/or star product philosophy, which are extensively applied within noncommutative quantum theories. Newton-Lagrange noncommutative equations of motion are formulated and their properties are analyzed from the pure geometrical point of view. It is argued that the dynamical quintessence of the system consists in its kinetic energy (Riemannian metric) specifying Riemann-Levi-Civita connection and thus the inertia geodesics of the free motion. Throughout the paper, ''noncommutativity'' is considered as an internal geometric structure of the configuration space, which can not be ''observed'' per se. Manifestation of the noncommutative phenomena is mediated by the interaction of the system with noncommutative background under the consideration. The simplest model of the interaction (minimal coupling) is proposed and it is shown that guiding affine connection is modified by the quadratic analog of the Lorentz electromagnetic force (contortion term).

Key words: noncommutative mechanics; affine connection; contortion.

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