  This paper presents symmetry reduction for material stochastic Lagrangian
systems with advected quantities whose configuration space is a Lie group. Such
variational principles yield deterministic as well as stochastic constrained
variational principles for dissipative equations of motion in spatial
representation. The general theory is presented for the finite dimensional
situation. In infinite dimensions we obtain partial differential equations and
stochastic partial differential equations. When the Lie group is, for example,
a diffeomorphism group, the general result is not directly applicable but the
setup and method suggest rigorous proofs valid in infinite dimensions which
lead to similar results. We apply this technique to the compressible
Navier-Stokes equation and to magnetohydrodynamics for charged viscous
compressible fluids. A stochastic Kelvin-Noether theorem is presented. We
derive, among others, the classical deterministic dissipative equations from
purely variational and stochastic principles, without any appeal to
thermodynamics.
