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Kaptsov, E. I.; Meleshko, S. V.

Analysis of the one-dimensional Euler-Lagrange equation of continuum mechanics with a Lagrangian of a special form. (English) Zbl 1481.76008

Appl. Math. Modelling 77, Part 2, 1497-1511 (2020).
Summary: The equations describing the flow of a one-dimensional continuum in Lagrangian coordinates are studied in this paper by the group analysis method. They are reduced to a single Euler-Lagrange equation which contains two undetermined functions (arbitrary elements). Particular choices of these arbitrary elements correspond to different forms of the shallow water equations, including those with both, a varying bottom and advective impulse transfer effect, and also some other motions of a continuum. A complete group classification of the equations with respect to the arbitrary elements is performed. One advantage of the Lagrangian coordinates consists of the presence of a Lagrangian, so that the equations studied become Euler-Lagrange equations. This allows us to apply Noether’s theorem for constructing conservation laws in Lagrangian coordinates. Not every conservation law in Lagrangian coordinates has a counterpart in Eulerian coordinates, whereas the converse is true. Using Noether’s theorem, conservation laws which can be obtained by the point symmetries are presented, and their analogs in Eulerian coordinates are given, where they exist.

Cited in 12 Documents

MSC:

76A02 Foundations of fluid mechanics
37J51 Action-minimizing orbits and measures for finite-dimensional Hamiltonian and Lagrangian systems; variational principles; degree-theoretic methods
70H03 Lagrange’s equations

Keywords:

Lagrangian fluid dynamics; Lie group; group classification; conservation law; Noether’s theorem
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References:

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