  The paper considers one-dimensional flows of a polytropic gas in the
Lagrangian coordinates in three cases: plain one-dimensional flows, radially
symmetric flows and spherically symmetric flows. The one-dimensional flow of a
polytropic gas is described by one second-order partial differential equation
in the Lagrangian variables. Lie group classification of this PDE is performed.
Its variational structure allows to construct conservation laws with the help
of Noether's theorem. These conservation laws are also recalculated for the gas
dynamics variables in the Lagrangian and Eulerian coordinates. Additionally,
invariant and conservative difference schemes are provided.
