Geodesic
On a~form of the first variation of the action integral over a~varied domain
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 199-209

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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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     author = {V. A. Kovalev and Yu. N. Radayev},
     title = {On a~form of the first variation of the action integral over a~varied domain},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {199--209},
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V. A. Kovalev; Yu. N. Radayev. On a~form of the first variation of the action integral over a~varied domain. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 199-209. http://geodesic.mathdoc.fr/item/ISU_2014_14_2_a8/