Group Analysis of the One-Dimensional Gas Dynamics Equations in Lagrangian Coordinates and Conservation Laws

A group analysis of the second-order equation including the one-dimensional gas dynamics equations in Lagrangian coordinates as a particular case is performed. The use of Lagrangian coordinates makes it possible to consider the one-dimensional gas dynamics equations as a variational Euler-Lagrange equation with an appropriate Lagrangian. Conservation laws are derived with the use of the variational presentation and Noether’s theorem. A complete group classification of the Euler-Lagrange equation is obtained; as a result, 18 different classes can be identified.