Symmetries and conservation laws of 2-dimensional ideal plasticity

Abstract

Symmetry theory is of fundamental importance in studying systems of partial differential equations. At present algebras of classical infinitesimal symmetry transformations are known for many equations of continuum mechanics [1, 2, 4]. Methods foi finding these algebras go back to S. Lie's works written about 100 years ago. Ir particular, knowledge of symmetry algebras makes it possible to construct effectively wide classes of exact solutions for equations under consideration and via Noether's theorem to find conservation laws for Euler–Lagrange equations. The natural development of Lie's theory is the theory of “higher” symmetries and conservation laws [5].

Symmetries and conservation laws of partial differential equations: Basic notions and results

“…Therefore, 1) is a conserved current for (1) as well as for every other equation posed on dependent variables u ~ ..... um. This shows that conserved currents of the form (14) are nonspecific for Equations (1) and, therefore, for the physical object described by them. In other words, they contain no meaningful information about this object.…”

“…An interested reader will cover this gap by consulting [4,13,17]. We also recommend paper [14] in which all the main technological details necessary for finding symmetries and conservation laws are demonstrated in a concrete example.…”

Solution of Boundary Value Problems of Plasticity with the Use of Conservation Laws

“…This quite a commonplace definition is reasonably suitable for the presentation of our results. More general approach can be learned in [11].…”

Application of conservation laws to Dirichlet problem for elliptic quasilinear systems

“…1), i.e. it is necessary to determine the points ( )∈ x y S , 0 0 , where the solution of Dirichlet problem (11), (13), and (14) satisfies condition (12). Note that along Γ condition (12) is satisfied.…”

“…There appear two principal directions of investigations. The first one is the mathematical description of conservation laws of a given system (for instance [13,14]) and the second one is using CL to determine solutions of PDEs in practical applications. There is a lot of results in both directions.…”