Specific features of differential equations of mathematical physics

Three types of equations of mathematical physics, namely, the equations, which describe any physical processes, the equations of mechanics and physics of continuous media, and field-theory equations are studied in this paper. In the first and second case the investigation is reduced to the analysis of the nonidentical relations of the skew-symmetric differential forms that are obtained from differential equations. It is shown that the integrability of equations and the properties of their solutions depend on the realization of the conditions of degenerate transformations under which the identical relations are obtained from the nonidentical relation. The field-theory equations, in contrast to the equations of first two types, are the relations made up by skew-symmetric differential forms or their analogs (differential or integral ones). This is due to the fact that the field-theory equations have to describe physical structures (to which closed exterior forms correspond) rather than physical quantities. The equations that correspond to field theories are obtained from the equations that describe the conservation laws (of energy, linear momentum, angular momentum, and mass) of material systems (of continuous media). This disclose a connection between field theories and the equations for material systems (and points to that material media generate physical fields). 1. Specific features of equations descriptive of physical processes Specific features of differential equations descriptive of physical processes and the types of their solutions can be demonstrated by the example of first-order partial differential equation using the properties of skew-symmetric differential forms. [The method of investigating differential equations using skew-symmetric differential forms was developed by Cartan [1] in his analysis of the integrability of differential equations. Here we present this analysis to demonstrate specific features of differential equations and properties of solutions to these equations.] Let F (x, u, pi) = 0, pi = ∂u/∂x i (1) be a first-order partial differential equation. Let us consider the functional relation