Abstract
One-dimensional motion of fluids in Lagrangian coordinates is considered in this study. The observation that the equa- tions of fluids with internal inertia in Lagrangian coordinates have the form of an Euler-Lagrange equation with a natural Lagrangian allows us to apply Noether’s theorem for constructing conservation laws. The complete group classification with respect to a state equation of the one-dimensional gas dynamics type equations in Lagrangian coordinates is obtained. Using Noether’s theorem, conservation laws in Lagrangian coordinates are constructed. For the hyperbolic shallow water equations and the Green-Naghdi equations conservation laws are found.
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