Nontrivial Critical Networks. Singularities of Lagrangians and a Criterion for Criticality

We single out the class of so-called quasiregular Lagrangians, which have singularities on the zero section of the cotangent bundle to the manifold on which extremal networks are considered. A criterion for a network to be extremal is proved for such Lagrangians: the Euler--Lagrange equations must be satisfied on each edge, and some matching conditions must be valid at the vertices.