Conservation laws of the equation of one-dimensional shallow water over uneven bottom in Lagrange's variables

The systems of equations of one-dimensional shallow water over uneven bottom in Euler's and Lagrange's variables are considered. The intermediate system of equations is introduced. Hydrodynamic conservation laws of intermediate system of equations are used to find all first order conservation laws of shallow water equations in Lagrange's variables for all bottom profiles. The obtained conservation laws are compared with the hydrodynamic conservation laws of the system of equations of one-dimensional shallow water over uneven bottom in Euler's variables. Bottom profiles, providing additional conservation laws, are given. The problem of group classification of contact transformations of the shallow water equation in Lagrange's variables is solved. First order conservation laws of the shallow water equation in Lagrange's variables are obtained using Noether's theorem. In the considered cases, the correspondence between non-divergence symmetries and the original Lagrangian is shown. A similar correspondence is valid for an arbitrary ordinary differential equation of the second order. It is shown that the application of Lagrange's identity did not find all first order conservation laws of the shallow water equation in Lagrange's variables.