We derive deterministic dissipative equations of motion from variational principles for random Lagrangian paths. These are ordinary differential equations when the configuration space is finite dimensional and partial differential equations in the infinite dimensional case. We consider systems with advected quantities, using semidirect product Lie algebras. Examples cover compressible Navier-Stokes equations and magnetohydrodynamics (MHD) for charged viscous compressible fluids. Finally, we can also consider stochastic differential equations and stochastic partial differential equations of dissipative systems by randomly perturbing the Lagrangian.
This is joint work with Xin Chen and Tudor Ratiu (Shanghai Jiao Tong Univ.)