Adjoint Variational Principles for Regularised Conservative Systems

Variational principles are powerful tools in many branches of theoretical physics. Certain conservative systems which do not admit of a traditional Euler-Lagrange variational formulation are given a novel generalization. Illustrative examples, including the recently discovered scale-invariant analogue of the Korteweg-de Vries equation are presented. The new "adjoint variational method" is applied to regularized, incompressible, conservative hydrodynamics expressed in Eulerian variables, as opposed to the usual Lagrangian variables. The regularized, two-fluid, non-dissipative, quasineutral, incompressible plasma equations [known as "Hall MHD" and the electromagnetic field equations are derived from the new formulation. It turns out that the associated adjoint equations are precisely the two-fluid "cross-helicity/frozen-field" theorems pertaining to these regularized systems which have no standard variational formulation). The adjoint equations also provide a direct route to the integral invariants of the system and suggest new analytical and numerical approaches to the dynamics