New Hamiltonian formalism and Lagrangian representations for integrable hydrodynamic type systems

TL;DR

This paper introduces a new Hamiltonian formalism for integrable hydrodynamic systems based on conjugate coordinate nets, providing mirrored formulas, Lagrangian formulations, and multiple Hamiltonian structures, including generalizations and examples.

Contribution

It develops a novel Hamiltonian formalism for hydrodynamic systems, extending existing theories with mirrored formulas, Lagrangian representations, and new Hamiltonian structures of various orders.

Findings

Established a new Hamiltonian formalism based on conjugate coordinate nets.

Derived mirrored formulas and Lagrangian formulations for the systems.

Presented multiple Hamiltonian structures, including generalizations and examples.

Abstract

New Hamiltonian formalism based on the theory of conjugate curvilinear coordinate nets is established. All formulas are ``mirrored'' to corresponding formulas in the Hamiltonian formalism constructed by B.A. Dubrovin and S.P. Novikov (in a flat case) and E.V. Ferapontov (in a non-flat case). In the ``mirrored-flat'' case the Lagrangian formulation is found. Multi-Hamiltonian examples are presented. In particular Egorov's case, generalizations of local Nutku--Olver's Hamiltonian structure and corresponding Sheftel--Teshukov's recursion operator are presented. An number of Hamiltonian structures of all odd orders is found.