Hydrodynamics, particle relabelling and relativity

TL;DR

This paper explores extending hydrodynamic models to second-order theories, demonstrating how wave equations can be represented in Eulerian and Lagrangian frameworks consistent with relativity principles.

Contribution

It introduces the concept of particle relabelling relativity and connects Lagrangian and Eulerian formulations through canonical transformations.

Findings

Representation of wave equations in Eulerian models with Lorentz invariance

Compatibility of trajectory models with the relativity principle

Derivation of Eulerian conserved charges from Lagrangian symmetries

Abstract

Using the wave equation as an example, it is shown how to extend the hydrodynamic Lagrangian-picture method of constructing field evolution using a continuum of trajectories to second-order theories. The wave equation is represented through Eulerian-picture models that are distinguished by their Lorentz transformation properties. Introducing the idea of the relativity of the particle label, it is demonstrated how the corresponding trajectory models are compatible with the relativity principle. It is also shown how the Eulerian variational formulation may be obtained by canonical transformation from the Lagrangian picture, and how symmetries in the Lagrangian picture may be used to generate Eulerian conserved charges.