The most widely used numerical method used in linear calculation of building structures is finite element method in traditional form of displacements. Different software is developed on its basis. Though it is only possible to check the certainty of these numerical solutions, especially of non-linear tasks of engineering structures’ deformation by the coincidence of the results obtained by two different methods. The authors solved geometrically nonlinear task of the static deformation of a flat hinged-rod system consisting of five linear elastic rods undergoing great tension-compression strains. The solution was obtained basing on the finite element method in the form of classical mixed method developed by the authors. The set of all equilibrium states of the system, both stable and unstable, and all the limit points were found. The certainty of the solution was approved by the coincidence of the results obtained by other authors basing on traditional finite element method in displacements.

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А proof of reciprocity relations for nonlinear systems in inhomogeneous variable electric and magnetic fields in the presence of unsteady spin currents, thermodynamic flows and mechanical disturbances is obtained by the Kubo method in the approximation of Markov relaxation and locally quasi-equilibrium distribution.

The quantum proof of the reciprocal relations for nonlinear non-stationary systems in a magnetic field is obtain in the approach of Markov’s relaxation.