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 Izv. Saratov Univ. Math. Mech. Inform., 2014, Volume 14, Issue 2, Pages 199–209 (Mi isu502)

Mechanics

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

V. A. Kovaleva, Yu. N. Radayevb

a Moscow City Government University of Management, 28, Sretenka str., 107045, Moscow, Russia
b Institute for Problems in Mechanics of RAS, 101-1, Vernadskogo ave., 119526, Moscow, Russia

Abstract: 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.

Key words: field, action, least action principle, field equations, transformation group, Lie group, infinitesimal generator, variation, varied domain, constraint.

DOI: https://doi.org/10.18500/1816-9791-2014-14-2-199-209

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UDC: 539.374

Citation: V. A. Kovalev, Yu. N. Radayev, “On a form of the first variation of the action integral over a varied domain”, Izv. Saratov Univ. Math. Mech. Inform., 14:2 (2014), 199–209

Citation in format AMSBIB
\Bibitem{KovRad14} \by V.~A.~Kovalev, Yu.~N.~Radayev \paper On a~form of the first variation of the action integral over a~varied domain \jour Izv. Saratov Univ. Math. Mech. Inform. \yr 2014 \vol 14 \issue 2 \pages 199--209 \mathnet{http://mi.mathnet.ru/isu502} \crossref{https://doi.org/10.18500/1816-9791-2014-14-2-199-209} \elib{https://elibrary.ru/item.asp?id=21719218} 

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This publication is cited in the following articles:
1. E. V. Murashkin, Yu. N. Radayev, “On a micropolar theory of growing solids”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 24:3 (2020), 424–444
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