One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: symmetry classification, conservation laws, difference schemes

Lie point symmetries of the one-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates are considered. Complete Lie group classification of these equations reduced to a scalar second-order PDE is performed. The classification parameter is the entropy. Noether theorem is applied for constructing conservation laws. The conservation laws can be represented in the gas dynamics variables. For the basic adiabatic case invariant and conservative difference schemes are discussed.