Lin constraints, Clebsch potentials and variational principles

Abstract

The Poisson bracket formulation of fluid, plasma and rigid body type systems has undergone considerable recent development using techniques of symmetry group reduction. The relationship between this approach and that using Lin constraints and Clebsch potentials is established. The connection is made in the setting of abstract Clebsch variables as well as that of variational principles on reduced spaces. Variational principles for both the Clebsch and reduced form (such as fluids in spatial representation) are derived from the standard variational principle of Hamilton in material (Lagrangian) representation using reduction theory.