Mathematical models and contemporary theories of physical fields

Elements of the classical field theory based on a variational formulation of the Hamilton type are discussed and corresponding 4-dimensional Lagrange formalism is presented both as the variational and the group theoretical script. Variational symmetries (geometric and generalized) of field equations and theNoether theoremproviding a regular way of obtaining a conservation law for every given variational symmetry are revisited in the study in order to give a complete version of the contemporary field theory. All developments are presented in the non-linear frame (i.e. of finite strains as to continuum mechanics). Natural derivations of all tensor attributes of a physical field are given by the variational symmetry technique. The null Lagrangian theory for $n$-dimensional manifold (including 4-dimensional Minkowski space-time) is developed in an attempt to extend the canonical formalismof non-linear field theory. By the aid of divergence formula for the null Lagrangians regular in $n$-dimensional star-shaped domains, a general representation of the null Lagrangian depending as maximum on the first order field gradients is obtained. A method of systematic and derivation of the null Lagrangians for $n$-dimensional manifold is proposed. It is shown that in the case of non-linear 3-component field in 3-dimensional space the null Lagrangian is represented, in general, via 15 arbitrary independent field functions.