Group analysis of one-dimensional gas dynamics equations in Lagrangian coordinates and conservation laws

A group analysis of the second-order equation including one-dimensional gas dynamics equations in Lagrangian coordinates as a particular case is performed. The use of Lagrangian coordinates makes it possible to consider 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 theory. A complete group classification of the Euler–Lagrange equation is obtained; as a result, 18 different classes can be classified.