Reciprocal transformations and local Hamiltonian structures of hydrodynamic type systems

TL;DR

This paper investigates conditions under which reciprocal transformations of hydrodynamic type systems preserve flatness of metrics, revealing that only specific linear conservation laws lead to flat reciprocal metrics and linking geometric equivalences to Hamiltonian structures.

Contribution

It characterizes when reciprocal transformations preserve flat or constant curvature metrics in hydrodynamic systems, connecting geometric and Hamiltonian structures.

Findings

Reciprocal metrics are flat only if conservation laws are linear combinations of canonical densities.

Classifies reciprocal transformations that preserve flatness in systems with flat or constant curvature initial metrics.

Shows geometric equivalence of systems corresponds to reciprocal transformations relating their Hamiltonian structures.

Abstract

We start from a hyperbolic DN hydrodynamic type system of dimension $n$ which possesses Riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that, after such a transformation of the independent variables, one of the metrics associated to the initial system be flat. We prove the following statement: let $n\ge 3$ in the case of reciprocal transformations of a single independent variable or $n\ge 5$ in the case of transformations of both the independent variable; then the reciprocal metric may be flat only if the conservation laws in the transformation are linear combinations of the canonical densities of conservation laws, {\it i.e} the Casimirs, the momentum and the Hamiltonian densities associated to the Hamiltonian operator for the initial metric. Then, we restrict ourselves to the case in which the initial metric is either flat or of constant curvature and we classify the reciprocal transformations of one or both the independent variables so that the reciprocal metric is flat. Such characterization has an interesting geometric interpretation: the hypersurfaces of two diagonalizable DN systems of dimension $n\ge 5$ are Lie equivalent if and only if the corresponding local hamiltonian structures are related by a canonical reciprocal transformation.