Cite this article as:
Kovalev V. A., Radaev Y. N. On a form of the first variation of the action integral over a varied domain . Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 2, pp. 199-209. DOI: https://doi.org/10.18500/1816-9791-2014-14-2-199-209
On a form of the first variation of the action integral over a varied domain
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.
1. Kovalev V. A., Radayev Yu. N. Elements of the Field Theory : Variational Symmetries and Geometric Invariants. Moscow, Fizmatlit, 2009, 156 p. (in Russian).
2. Kovalev V. A., Radayev Yu. N. Wave Problems of Field Theory and Thermomechanics. Saratov, Saratov Univ. Press, 2010. 328 p. (in Russian).
3. Ovsyannikov L. V. Group Analysis of Differential Equations. Moscow, Nauka, 1978, 400 p. (in Russian).
4. Olver P. J. Applications of Lie Groups to Differential Equations. Moscow, Mir, 1989, 639 p. (in Russian).
5. Olver P. J. Equivalence, Invariants and Symmetry. Cambridge ; N.Y. ; Melbourne, Cambridge University Press, 1995, 526 p.
6. Noether E. Invariante Variationsprobleme. Nachrichten von der K¨oniglichen Gesellschaft der Wissenschaften zu G¨ottingen. Mathematisch-Physikalische Klasse,
1918, H. 2, s. 235–257. (Rus. ed. : Noether E. Invariantnye variatsionnye zadachi // Variatsionnye printsipy mekhaniki. Moscow, Fizmatgiz, 1959, pp. 611–630).
7. Courant R., Gilbert D. Methods of Mathematical Physics : in 2 vol. Moscow; Leningrad, Gostekhteoretizdat, 1933, vol. 1, 528 p. (in Russian).
8. Gelfand I. M., Fomin S. V. Calculus of Variations. Moscow, Fizmatgiz, 1961, 228 p. (in Russian).
9. Gunter N. M. A Cours of the Calculus of Variations. Moscow; Leningrad, Gostekhteoretizdat, 1941, 308 p. (in Russian).
