Abstract
Within the framework of the Lagrangian approach, a mathematically correct procedure is given for obtaining the necessary conditions of locextr from the variational principle for a number of continuous media that fill a certain geometric region at each moment of time and does not leave it over time. The proposed method is based on the geometric properties of the set of diffeomorphisms of the indicated domain, which is the configuration space of a continuous medium, and on a certain embedding theorem. The efficiency of the method is demonstrated by examples of obtaining the necessary locextr conditions from variational principles for ideal gas dynamics and classical MHD plasma. The central result of the work is the construction of the variational principle, in particular, the Lagrangian and the action functional for the theory of electromagnetic hydrodynamics of plasma.
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