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Flows of one-dimensional continuum in Lagrangian coordinates are studied inthe paper. Equations describing these flows are reduced to a singleEuler-Lagrange equation which contains two undefined functions. Particularchoices of the undefined functions correspond to isentropic flows of an idealgas, different forms of the hyperbolic shallow water equations. Complete groupclassification of the equation with respect to these functions is performed.Using Noether's theorem, all conservation laws are obtained. Their analogs inEulerian coordinates are given.

Analysis of the One-dimensional Euler-Lagrange equation of continuum mechanics with a Lagrangian of a special form