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We give the realizations by sequences of ordinary (almost everywhere) differential equations of the implicit random exact representations for the Navier-Stokes equations on smooth compact manifolds isometrically immersed in Euclidean spaces (viz. spheres, tori, euclidean spaces, etc.). We construct the random Hamiltonean system structure of the Navier-Stokes equations using the extensions of these realizations to the cotangent manifold and their associated classical Hamiltonean structure.This construction converges on probability to an associated random symplectic structure which turns to be the canonical one, albeit the coordinates in phase-space are random continuous functions. We obtain a Liouville measure for the Navier-Stokes equations, and construct a random Poincaré-Cartan 1-form. In the case of vanishing kinematical viscosity, we obtain the Arnold-Ebin-Marsden symplectic theory for incompressible perfect fluids obeying the Euler equations. Finally we discuss the relation between this approach and the low-dimensional approach to turbulence by classical dynamical systems.

Random symplectic geometry and the realizations of the random representations of the Navier-Stokes equations by ordinary differential equations