Symmetry in Nonlinear Mathematical Physics - 2009

Roman Popovych (Institute of Mathematics, Kyiv, Ukraine & University of Vienna, Austria)

Potential conservation laws

We prove that in the case of two independent variables local conservation laws of potential systems have characteristics depending only on local variables if and only if they are induced by local conservation laws of the corresponding initial systems of differential equations. Therefore, characteristics of pure potential conservation laws have to essentially depend on potential variables that gives a criterion for the selection of such conservation laws. Moreover, we present extensions to multi-dimensional standard and gauged potential systems, Abelian and general coverings and general foliated systems of differential equations. An example illustrating possible applications of these results is given. A special version of the Hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws.