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IssuesArchive of Issues2012-1pp.137-154

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V.A. Kovalev and Yu.N. Radaev, "Forms of Null Lagrangians in Field Theories of Continuum Mechanics," Mech. Solids. 47 (1), 137-154 (2012)
Year 2012 Volume 47 Number 1 Pages 137-154
DOI 10.3103/S002565441201013X
Title Forms of Null Lagrangians in Field Theories of Continuum Mechanics
Author(s) V.A. Kovalev (Moscow City Government University of Management, Sretenka 28, Moscow, 107045 Russia, vlad_koval@mail.ru)
Yu.N. Radaev (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526 Russia, radayev@ipmnet.ru, y.radayev@gmail.com)
Abstract The divergence representation of a null Lagrangian that is regular in a star-shaped domain is used to obtain its general expression containing field gradients of order ≤1 in the case of spacetime of arbitrary dimension. It is shown that for a static three-component field in the three-dimensional space, a null Lagrangian can contain up to 15 independent elements in total. The general form of a null Lagrangian in the four-dimensional Minkowski spacetime is obtained (the number of physical field variables is assumed arbitrary). A complete theory of the null Lagrangian for the n-dimensional spacetime manifold (including the four-dimensional Minkowski spacetime as a special case) is given. Null Lagrangians are then used as a basis for solving an important variational problem of an integrating factor. This problem involves searching for factors that depend on the spacetime variables, field variables, and their gradients and, for a given system of partial differential equations, ensure the equality between the scalar product of a vector multiplier by the system vector and some divergence expression for arbitrary field variables and, hence, allow one to formulate a divergence conservation law on solutions to the system.
Keywords symmetry, field gradients, null Lagrangians, physical field variables, variational calculus, divergence conservation law
References
1.  P. J. Olver, Applications of Lie Groups to Differential Equations (Mir, Moscow, 1989) [in Russian].
2.  P. J. Olver, Equivalence, Invariants, and Symmetry (Cambridge Univ. Press, Cambridge, etc., 1995).
3.  M. Silhavy, The Mechanics and Thermodynamics of Continuous Media (Springer, Berlin, 1997).
4.  Yu. N. Radaev and V. A. Gudkov, "On Calculation of Null Lagrangians of a Nonlinear Elastic Field," Vestn. Samarsk. Gos. Univ. Estestv. Ser., Special issue No. 4(32), 39-56 (2002).
5.  R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Gostekhizdat, Moscow-Leningrad, 1933) [in Russian].
6.  L. Schwartz, Analysis, Vol. 2 (Mir, Moscow, 1972) [in Russian].
7.  H. Cartan, Differential Calculus. Differential Forms (Mir, Moscow, 1971) [in Russian].
8.  A. J. McConnell, Introduction to Tensor Analysis with Applications to Geometry, Mechanics, and Physics (Fizmatgiz, Moscow, 1963) [in Russian].
Received 19 November 2009
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