Abstract
By rigorous analysis, it is proven that from discontinuous Lagrangians, which are invariant with respect to the Galilean group, Rankine–Hugoniot conditions for propagating discontinuities can be derived via a straight forward procedure that can be considered an extension of Noether’s theorem. The use of this general procedure is demonstrated in particular for a Lagrangian for viscous flow, reproducing the well known Rankine–Hugoniot conditions for shock waves.
Highlights
The formulation of physical theories within the framework of variational principles enables a deeper understanding of the physical system in many respects [1]
Such considerations are outside the Lagrangian formalism, the Rankine–Hugoniot conditions are essentially based on conservation of physical quantities, such as mass, momentum and energy
Based on a consistent analysis of the Galilean group and its implications for the structure of Lagrangians, as well as preliminary work with respect to dealing with discontinuous Lagrangian densities, it was possible to transfer the concept that every Lie-type symmetry can be assigned a physical conservation law, originally formulated by E
Summary
The formulation of physical theories within the framework of variational principles enables a deeper understanding of the physical system in many respects [1]. Analogous conditions are used in fracture mechanics [6] Such considerations are outside the Lagrangian formalism, the Rankine–Hugoniot conditions are essentially based on conservation of physical quantities, such as mass, momentum and energy. This poses the question of whether there is a way to obtain such conditions via Noether’s theorem, applied to fundamental symmetries of the Lagrangian and, in particular, the Galilean group. On introduction of the complex field χ, a discontinuity is imposed in the resulting Lagrangian, which goes beyond the usual framework in continuum theory so that from the variation with respect to the fundamental fields, in addition to the well-known Euler–Lagrange equations, corresponding transition and production conditions along the discontinuity surfaces result. The so-called Euler–Lagrange equations [1]
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