#### For citation:

Kovalev V. A., Radaev Y. N. On a form of the first variation of the action integral over a varied domain. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2014, vol. 14, iss. 2, pp. 199-209. DOI: 10.18500/1816-9791-2014-14-2-199-209

# On a form of the first variation of the action integral over a varied domain

Field theories of the continuum mechanics and physics based on the least action principle are considered in a unified framework. Variation of the action integral in the least action principle corresponds variations of physical fields while space-time coordinates are not varied. However notion of the action invariance, theory of variational symmetries of action and conservation laws require a wider variation procedure including variations of the space-time coordinates. A similar situation is concerned to variational problems with strong discontinuities of field variables or other a priori unknown free boundaries which variations are not prohibited from the beginning. A form of the first variation of the action integral corresponding variations of space-time coordinates and field variables under one-parametrical transformations groups is obtained. This form is attributed to 4-dimensional covariant formulations of field theories of the continuum mechanics and physics. The first variation of the action integral over a varied domain is given for problems with constraints. The latter are formulated on unknown free boundaries.

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