  Poincare's invariance principle for Hamiltonian flows implies Kelvin's
principle for solution to Incompressible Euler Equation. Iyer-Constantin
Circulation Theorem offers a stochastic analog of Kelvin's principle for
Navier-Stokes Equation. Weakly symplectic diffusions are defined to produce
stochastically symplectic flows in a systematic way. With the aid of symplectic
diffusions, we produce a family of martigales associated with solutions to
Navier-Stokes Equation that in turn can be used to prove Iyer-Constantin
Circulation Theorem. We also review some basic facts in symplectic and contact
geometry and their applications to Euler Equation.
