Abstract
Differential-geometry structures associated with Lagrangians are studied. A relative invariant E embraced by an extension of fundamental object is constructed (in the paper, E is referred to as the Euler relative invariant) such that the equation E = 0 is an invariant representation of the Euler equation for the variational functional. For this reason, a nonvariational interpretation of the Euler equations becomes possible, because the Euler equations need not be connected with the variational problem, and one can regard the equations from the very beginning as an equation arising when equating the Euler relative invariant to zero.
Local diffeomorphisms between two structures associated with Lagrangians are also discussed. The theorem concerning conditions under which the vanishing condition for the Euler relative invariant of one of these structures leads to vanishing of the Euler invariant relative of the other structure can be treated as a nonvariational interpretation of Nöther’s theorem.
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References
A. M. Vasil’ev, Theory of Differential-Geometric Structures (Moskov. Gos. Univ., Moscow, 1987) [in Russian].
G. F. Laptev, “Fundamental Infinitesimal Structures of Higher Orders on a Smooth Manifold,” Tr. Geometr. Sem. 1 (VINITI, Moscow, 1966), pp. 139–189 [in Russian].
M. V. Vasil’eva, “Geometry of an integral,” Mat. Sb. N.S. 36(7) (1), 57–92 (1955) [in Russian].
L. E. Evtushik, “On the Geometry of a Double Integral,” Mat. Sb. N.S. 37(79)(1), 197–208 (1955) [in Russian].
L. E. Evtushik, Ju. G. Lumiste, N. M. Ostianu, and A. P. Shirokov, “Differential-Geometric Structures on Manifolds,” in: Problems in geometry, Vol. 9 (Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979), pp. 5–246 [in Russian].
G. F. Laptev, “Differential Geometry of Imbedded Manifolds. Group Theoretical Method of Differential Geometric Investigations,” Tr. Moskov. Mat. Obs. 2, 275–382 (1953) [in Russian].
G. F. Laptev, “Group Theoretical Method of Differential Geometric Investigations,” in: Proc. Third All-Union Math. Congr. Moscow 3, (1956) (AN SSSR; Moscow 409–418, 1958) [in Russian].
G. F. Laptev, “The Structural Equations of a Principal Bundle Manifold,” Tr. Geometr. Sem. 2 (VINITI, Moscow, 1969), pp. 161–178 [in Russian].
L. É. El’sgol’ts, Differential Equations and Calculus of Variations,, Course in Higher Mathematics and Mathematical Physics, No. 3 (Izdat. “Nauka,” Moscow, 1965) [in Russian].
A.K. Rybnikov, “Differential-Geometric Structures Defining Lie-Bäcklund Transformations,” Dokl. Akad. Nauk Ross. Akad. Nauk 425(1), 25–30 (2009) [Dokl. Math. 79 (2), 163–168 (2009)].
A. K. Rybnikov, “Differential-Geometric Structures Defining Higher Order Contact Transformations,” Izv. Vyssh. Uchebn. Zaved. Mat. (9), 70–89 (2011) [Russ. Math. 55 (9), 58–75 (2011)].
G. F. Laptev, “On the Invariant Analytic Theory of Differentiable Mappings,” Tr. Geometr. Sem. 6 (VINITI, Moscow, 1974), pp. 37–42 [in Russian].
I. M. Gel’fand and S. V. Fomin, Variational Calculus (Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961) [in Russian].
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Rybnikov, A.K. Differential-geometric structures associated with the Lagrangian and a nonvariational interpretation of the Euler equations and Nöther’s theorem. Russ. J. Math. Phys. 20, 336–344 (2013). https://doi.org/10.1134/S1061920813030084
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DOI: https://doi.org/10.1134/S1061920813030084