Abstract
We propose a new approach in Lagrangian formalism for studying the fluid dynamics on noncommutative space. Starting with the Poisson bracket for single particle, a map from canonical Lagrangian variables to Eulerian variables is constructed for taking into account the noncommutative effects. The advantage of this approach is that the kinematic and potential energies in the Lagrangian formalism continuously change in the infinite limit to the ones in Eulerian formalism and hence make sure that both the kinematical and potential energies are taken into account correctly. Furthermore, in our approach, the equations of motion of the mass density and current density are naturally expressed into conservative form. Based on this approach, the noncommutative Poisson bracket is introduced, and the noncommutative algebra among Eulerian variables and the noncommutative corrections on the equations of motion are obtained. We find that the noncommutative corrections generally depend on the derivatives of potential under consideration. Furthermore, we find that the noncommutative algebra does modify the usual Friedmann equation, and the noncommutative corrections measure the symmetry properties of the density function ρ(z→) under rotation around the direction θ→. This characterization results in vanishing corrections for spherically symmetric mass density distribution and potential.
Highlights
It is a general recognition in modern physics that nontrivial geometry of the background spacetime can affect the dynamics of particles living in it
The paper is organized as follows: in Section 2, based on the single particle picture, we introduce our map from Lagrangian variables to Eulerian variables and briefly discuss its relation to the approach in [50]; in Section 3, we show how the noncommutative algebra can be taken into account via the approach given in Section 2; we focus on the noncommutative effects of the external potential; in Section 4, the cosmological implications of the noncommutative effects in our approach are discussed along the line in [50]; summary and our conclusions are given in the final section, Section 5
In summary we proposed a refined approach for studying the fluid dynamics on noncommutative space
Summary
It is a general recognition in modern physics that nontrivial geometry of the background spacetime can affect the dynamics of particles living in it. The bracket algebra between the canonical Lagrangian variables is extended in noncommutative space, and a map [55] from the Lagrangian variables to Eulerian variables is applied to obtain the noncommutative corrections in the fluid field theory. The paper is organized as follows: in Section 2, based on the single particle picture, we introduce our map from Lagrangian variables to Eulerian variables and briefly discuss its relation to the approach in [50]; in Section 3, we show how the noncommutative algebra can be taken into account via the approach given in Section 2; we focus on the noncommutative effects of the external potential; in Section 4, the cosmological implications of the noncommutative effects in our approach are discussed along the line in [50]; summary and our conclusions are given, Section 5 The paper is organized as follows: in Section 2, based on the single particle picture, we introduce our map from Lagrangian variables to Eulerian variables and briefly discuss its relation to the approach in [50]; in Section 3, we show how the noncommutative algebra can be taken into account via the approach given in Section 2; we focus on the noncommutative effects of the external potential; in Section 4, the cosmological implications of the noncommutative effects in our approach are discussed along the line in [50]; summary and our conclusions are given in the final section, Section 5
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