Generalized Noether’s theorem in classical field theory with variable mass

In this paper, the generalized Noether’s theorem for mechanical systems is extended to the classical fields with variable mass, i.e., to the corresponding continuous systems. Noether’s theorem is based on the modified Lagrangian, which, besides time derivatives of the field function, contains its partial derivatives with respect to the space coordinates. The generalized Noether’s theorem for the classical fields systems with variable mass enables us to find transformations of field functions and independent variables for which there are some integrals of motion. In the paper, Noether’s theorem is adopted for non-conservative fields, and energy integrals in a broader sense are determined. In the case of non-conservative fields, a complementary approach to the problem is introduced by applying so-called pseudo-conservative fields. It has been demonstrated that the pseudo-conservative systems have the same energy laws as the non-conservative fields where the laws are obtained by means of this generalized Noether’s theorem. As the special case, the natural classical fields with standard Lagrangian are considered.