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We study the effect of the topology of universe by gauging thenon-relativistic particle model on the torus and 3-torus, using the symplecticformalism of constrained systems and embedding those models on extendedphase-spaces. Also, we obtain the generators of the gauge transformations forgauged models. Extracting the corresponding Poisson structure of the existedconstraints, we show the effect of the topology on the canonical structure ofthe phase-spaces of those models and suggest some phenomenology to prove thetopology of the universe and probable non-commutative structure of the space.In addition, we show that the number of large extra dimensions in thePhase-spaces of the gauged embeded models are exactly two. Moreover, in theclassical form, we talk over MOND theory in order to study the origin of theterms appeared in the gauged theory, which modify the Newton's second law.

Lagrange multiplier and Wess-Zumino variable as large extra dimensions in the torus universe