Analysis of the One-dimensional Euler-Lagrange equation of continuum mechanics with a Lagrangian of a special form

TL;DR

This paper studies one-dimensional continuum flows in Lagrangian coordinates, reducing the governing equations to a single Euler-Lagrange form, classifying symmetries, and deriving conservation laws applicable in Eulerian coordinates.

Contribution

It provides a complete group classification of the Euler-Lagrange equation with respect to specific functions and derives all conservation laws using Noether's theorem.

Findings

Classification of flows including isentropic ideal gas and shallow water equations

Derivation of all conservation laws via Noether's theorem

Analogs of conservation laws in Eulerian coordinates

Abstract

Flows of one-dimensional continuum in Lagrangian coordinates are studied in the paper. Equations describing these flows are reduced to a single Euler-Lagrange equation which contains two undefined functions. Particular choices of the undefined functions correspond to isentropic flows of an ideal gas, different forms of the hyperbolic shallow water equations. Complete group classification of the equation with respect to these functions is performed. Using Noether's theorem, all conservation laws are obtained. Their analogs in Eulerian coordinates are given.