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Hydrodynamics on non-commutative space --A step toward hydrodynamics of granular materials--

Hydrodynamics on non-commutative space is studied based on a formulation of hydrodynamics by Y. Nambu in terms of Poisson and Nambu brackets. Replacing these brackets by Moyal brackets with a parameter $θ$, a new hydrodynamics on non-commutative space is derived. It may be a step toward to find the hydrodynamics of granular materials whose minimum volume is given by $θ$. To clarify this minimum volume, path integral quantization and uncertainty relation of Nambu dynamics are examined.