On the Various Aspects of Hamiltonian Description of the Mechanics of Continuous Media

TL;DR

This paper develops a Hamiltonian framework for continuous media starting from Lagrangian formalism, exploring symmetries, and applying it to plasma and gravitating gas, linking Euler and Lagrange descriptions.

Contribution

It introduces a Hamiltonian formalism for continuous media that incorporates volume-preserving symmetries and connects Euler and Lagrange descriptions, with applications to plasma and gravitating gas.

Findings

Relation of Thompson theorem to volume-preserving diffeomorphisms clarified

Euler and Lagrange descriptions are connected through a new $C^2$ formulation

Applications to plasma and gravitating gas demonstrate the framework's versatility

Abstract

We consider a general approach to the theory of continuous media starting from Lagrangian formalism. This formalism which uses the trajectories if constituents of media is very convenient for taking into account different types of interaction between particles typical for different media. Building the Hamiltonian formalism we discuss some issues which is not very well known, such as relation of famous Thompson theorem with the symmetry with respect to volume preserving diffeomorphisms. We also discuss the relation between Euler and Lagrange description and present similar to Euler $C^2$ formulation of continuous mechanics. In these general frameworks we consider as examples the theory of plasma and gravitating gas.