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

Abstract

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.

Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates

“…In [22] the group classification of the one-dimensional Euler-Lagrange equations of continuum mechanics was carried out by the authors. In particular, it was shown that the modified shallow water equation ( 7) is the Euler-Lagrange equation for the Lagrangian…”

“…One can find a more detailed discussion of the symmetries and conservation laws of the modified shallow water equations with different bottom topographies in [22].…”

“…The group properties of the shallow water equations were studied in numerous papers (see [17][18][19][20]). Group classification and first integrals of these equations can be found in [21,22]. It was shown (see, e. g., [22,23]) that the shallow water equations in Lagrangian coordinates can be obtained as Euler-Lagrange equations of Lagrangian functions of a special kind.…”

“…Group classification and first integrals of these equations can be found in [21,22]. It was shown (see, e. g., [22,23]) that the shallow water equations in Lagrangian coordinates can be obtained as Euler-Lagrange equations of Lagrangian functions of a special kind. Some exact solutions also can be found in [5,24,25].…”

Invariant solutions of the two-dimensional shallow water equations with a particular class of bottoms

“…Group properties of the two dimensional gas dynamics equations of a polytropic gas in mass Lagrangian coordinates were analyzed in [4]. It should be also noted the paper [6], where flows of one-dimensional continuum in Lagrangian coordinates are considered. Equations describing these flows are reduced to a single Euler-Lagrange equation which contains two undefined functions.…”