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This paper develops the theory of affine Euler-Poincar\'e and affineLie-Poisson reductions and applies these processes to various examples ofcomplex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluidsand superfluids, spin glasses, microfluids, and liquid crystals. As aconsequence of the Lagrangian approach, the variational formulation of theequations is determined. On the Hamiltonian side, the associated Poissonbrackets are obtained by reduction of a canonical cotangent bundle. AKelvin-Noether circulation theorem is presented and is applied to theseexamples.

The Geometric Structure of Complex Fluids