Constrained and stochastic variational principles for dissipative equations with advected quantities

Abstract

We develop symmetry reduction for material stochastic Lagrangian systems with advected quantities whose configuration space is a Lie group, obtaining deterministic constrained variational principles and dissipative equations of motion in spatial representation. We discuss in detail the situation when the group in the general theory is a group of diffeomorphisms and derive, as an application, an MHD system for viscous compressible fluids.