A practical form of Lagrange–Hamilton theory for ideal fluids and plasmas

Lagrangian and Hamiltonian formalisms for ideal fluids and plasmas have, during the last few decades, developed much in theory and applications. The recent formulation of ideal fluid/plasma dynamics in terms of the Euler–Poincaré equations makes a self-contained, but mathematically elementary, form of Lagrange–Hamilton theory possible. The starting point is Hamilton's principle. The main goal is to present Lagrange–Hamilton theory in a way that simplifies its applications within usual fluid and plasma theory so that we can use standard vector analysis and standard Eulerian fluid variables. The formalisms of differential geometry, Lie group theory and dual spaces are avoided and so is the use of Lagrangian fluid variables or Clebsch potentials. In the formal ‘axiomatic’ setting of Lagrange–Hamilton theory the concepts of Lie algebra and Hilbert space are used, but only in an elementary way. The formalism is manifestly non-canonical, but the analogy with usual classical mechanics is striking. The Lie derivative is a most convenient tool when the abstract Lagrange–Hamilton formalism is applied to concrete fluid/plasma models. This directional/dynamical derivative is usually defined within differential geometry. However, following the goals of this paper, we choose to define Lie derivatives within standard vector analysis instead (in terms of the directional field and the div, grad and curl operators). Basic identities for the Lie derivatives, necessary for using them effectively in vector calculus and Lagrange–Hamilton theory, are included. Various dynamical invariants, valid for classes of fluid and plasma models (including both compressible and incompressible ideal magnetohydrodynamics), are given simple and straightforward derivations thanks to the Lie derivative calculus. We also consider non-canonical Poisson brackets and derive, in particular, an explicit result for incompressible and inhomogeneous flows.