The Hamiltonian Formalism in Fluid Dynamics

We propose a Hamiltonian approach to fluid mechanics based on the dynamics formulated in terms of Lagrangian variables. The construction of the canonical variables of the fluid elucidates the origin of the Clebsch variables, introduced in the 19th century. The developed formalism permits relating the circulation conservation law (Thompson theorem) to the invariance of the theory under special diffeomorphisms and also establishing new conservation laws. We also discuss the difference between the Eulerian and the Lagrangian descriptions, pointing out the incompleteness of the former. The constructed formalism is also applicable to an ideal plasma. We conclude with several remarks about quantizing the fluid.