Geodesic
Extremals of the Dirichlet functional on the manifold of two-dimensional spheroids in SO(3)
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 3, Tome 192 (2021), pp. 94-101.

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An approach to the approximate calculation of extremals of the Dirichlet functional on the Banach Lie group of “singular spheroids” defined in an annulus on the coordinate plane centered at the origin is presented. The technique presented is based on the use of a special simplification (functional reduction) of the equation of extremals. It is shown that the functional reduction allows one to obtain a lower estimate for the Morse index of extremals. We use smooth Fredholm functionals with continuous symmetries on smooth Banach manifolds with Riemannian (Hilbert) framings.
Keywords: smooth functional, extremal, continuous symmetry, codifferential, reduction. 
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     title = {Extremals of the {Dirichlet} functional on the manifold of two-dimensional spheroids in {SO(3)}},
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T. Yu. Sapronova; S. L. Tsarev. Extremals of the Dirichlet functional on the manifold of two-dimensional spheroids in SO(3). Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 3, Tome 192 (2021), pp. 94-101. http://geodesic.mathdoc.fr/item/INTO_2021_192_a9/

[1] Atya M., Geometriya i fizika uzlov, Mir, M., 1995

[2] Darinskii B. M., Sapronov Yu. I., Tsarev S. L., “Bifurkatsii ekstremalei fredgolmovykh funktsionalov”, Sovr. mat. Fundam. napr., 12 (2004), 3–140

[3] Dolzhenkov A. A., Sapronov Yu. I., Sapronova T. Yu., “Nelokalnyi bifurkatsionnyi analiz nekotorykh konfiguratsii kirkhgofova sterzhnya”, Matematicheskie modeli i operatornye uravneniya, VGU, Voronezh, 2009, 27–41

[4] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya: metody i prilozheniya, Nauka, M., 1986

[5] Zaitsev V. F., Polyanin A. D., Spravochnik po obyknovennym differentsialnym uravneniyam. Tochnye resheniya, Fizmatlit, M., 1995

[6] Monastyrskii M. I., Topologiya kalibrovochnykh polei i kondensirovannykh sred, PAIMS, M., 1995

[7] Monopoli: Topologicheskie i variatsionnye metody, Mir, M., 1989

[8] Sapronov Yu. I., “Konechnomernye reduktsii v gladkikh ekstremalnykh zadachakh”, Usp. mat. nauk., 51:1 (1996), 101–132

[9] Sapronova T. Yu., “Bifurkatsii ekstremalei iz tochek kriticheskoi orbity pri razrushenii nepreryvnoi simmetrii”, Tr. mat. fak-ta VGU. Nov. ser., 4 (1999), 101–107

[10] Sapronova T. Yu., “O metode kvaziinvariantnykh podmnogoobrazii v teorii fredgolmovykh funktsionalov”, Topologicheskie metody nelineinogo analiza, VGU, Voronezh, 2000, 107–124

[11] Sapronova T. Yu., “O razrushenii kompaktnykh kriticheskikh orbit invariantnykh fredgolmovykh funktsionalov pri nesimmetrichnykh vozmuscheniyakh”, Tr. mat. fak-ta VGU. Nov. ser., 2 (18) (1997), 54–58

[12] Fomenko A. T., Fuks D. B., Kurs gomotopicheskoi topologii, Nauka, M., 1989