  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.
