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We propose a new approach in Lagrangian formalism for studying the fluiddynamics on noncommutative space. Starting with the Poisson bracket for singleparticle, a map from canonical Lagrangian variables to Eulerian variables isconstructed for taking into account of the noncommutative effects. Theadvantage of this approach is that the kinematic and potential energies in theLagrangian formalism continuously change in the infinite limit to the ones inEulerian formalism, and hence make sure that both the kinematical and potentialenergies are taken into account correctly. Furthermore, in our approach, theequations of motion of the mass density and current density are naturallyexpressed into conservative form. Based on this approach, the noncommutativePoisson bracket is introduced, and the noncommutative algebra among Eulerianvariables, as well as the noncommutative corrections on the equations of motionare obtained. We find that the noncommutative corrections generally depend onthe derivatives of potential under consideration. Furthermore, we find that thenoncommutative algebra does modify the usual Friedmann equation, and thenoncommutative corrections measure the symmetry properties of the densityfunction $\rho(\vec{z})$ under rotation around the direction $\vec{\theta}$.This characterization results in vanishing corrections for sphericallysymmetric mass density distribution and potential.

Noncommutative effects on the fluid dynamics and modifications of the Freidmann equation