Forms of null Lagrangians in field theories of continuum mechanics |
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Authors: | V A Kovalev Yu N Radaev |
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Institution: | 1.Moscow City Government University of Management,Moscow,Russia;2.Ishlinsky Institute for Problems in Mechanics,Russian Academy of Sciences,Moscow,Russia |
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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. |
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Keywords: | |
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