The Hamiltonian formalism in fluid dynamics. (English) Zbl 1178.76389

Antoniou, I.; Pronko, G. P.

Theor. Math. Phys. 141, No. 3, 1670-1685 (2004); translation from Teor. Mat. Fiz. 141, No. 3, 392-410 (2004).

Summary: 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.

MSC:

76X05	Ionized gas flow in electromagnetic fields; plasmic flow
82D10	Statistical mechanical studies of plasmas
76A02	Foundations of fluid mechanics
37K05	Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)

Keywords:

hydrodynamics; canonical variables; plasma