 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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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 |
| Link to Fulltext |
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