Topological atlas of the Kovalevskaya top in a~double field

M. P. Kharlamov ; P. E. Ryabov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 185-230.

Résumé

The article contains the rough topological analysis of the completely integrable system with three degrees of freedom corresponding to the motion of the Kovalevskaya top in a double field. This system is not reducible to a family of systems with two degrees of freedom. We introduce the notion of a topological atlas of an irreducible system. For the Kovalevskaya top in a double field, we complete the topological analysis of all critical subsystems with two degrees of freedom and calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the initial momentum map pointing out all chambers, families of $3$-tori, and $4$-atoms of their bifurcations. Basing on the ideas of A. T. Fomenko, we define the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all nondegenerate rank $1$ singularities.

Bibliographie

[1] Bolsinov A. V., Kozlov V. V., Fomenko A. T., “Printsip Mopertyui i geodezicheskie potoki na sfere, voznikayuschie iz integriruemykh sluchaev dinamiki tvërdogo tela”, UMN, 50:3 (1995), 3–32 | MR | Zbl

[2] Bolsinov A. V., Matveev S. V., Fomenko A. T., “Topologicheskaya klassifikatsiya integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody. Spisok sistem maloi slozhnosti”, UMN, 45:2 (1990), 49–77 | MR | Zbl

[3] Bolsinov A. V., Rikhter P., Fomenko A. T., “Metod krugovykh molekul i topologiya volchka Kovalevskoi”, Matem. sb., 191:2 (2000), 3–42 | DOI | MR | Zbl

[4] Bolsinov A. V., Fomenko A. T., “Integriruemye geodezicheskie potoki na sfere, porozhdënnye sistemami Goryacheva–Chaplygina i Kovalevskoi v dinamike tvërdogo tela”, Matem. zametki, 56:2 (1994), 139–142 | MR | Zbl

[5] Bolsinov A. V., Fomenko A. T., Integriruemye gamiltonovy sistemy. Geometriya, topologiya, klassifikatsiya, V 2-kh t., Izd. dom “Udmurtskii universitet”, Izhevsk, 1999 | MR

[6] Brailov Yu. A., “Topologiya bifurkatsionnykh diagramm integriruemykh sistem na poluprostykh algebrakh Li”, Dokl. Akad. nauk, 375:2 (2000), 151–153 | MR | Zbl

[7] Zotev D. B., “Fazovaya topologiya volchka Kovalevskoi v $\mathrm{SO}(2)$-simmetrichnom dvoinom silovom pole”, Mekh. tvërd. tela, 2004, no. 34, 66–71 | MR

[8] Zotev D. B., “Fazovaya topologiya 1-go klassa Appelrota volchka Kovalevskoi v magnitnom pole”, Fundament. i prikl. matem., 12:1 (2006), 95–128 | MR | Zbl

[9] Zung N. T., “Topologicheskie invarianty integriruemykh geodezicheskikh potokov na mnogomernom tore i sfere”, Tr. MIAN, 205, 1994, 73–90 | MR | Zbl

[10] Kuznetsov A. V., “Postroenie invarianta algebraicheskogo analoga periodicheskoi tsepochki Toda”, UMN, 47:5(287) (1992), 181–182 | MR | Zbl

[11] Lerman L. M., Umanskii Ya. L., “O klassifikatsii chetyrëkhmernykh integriruemykh gamiltonovykh sistem v rasshirennykh okrestnostyakh prostykh osobykh tochek”, Metody kachestvennoi teorii i teorii bifurkatsii, Gork. gos. un-t, Gorkii, 1988, 67–76

[12] Oshemkov A. A., “Klassifikatsiya giperbolicheskikh osobennostei ranga nul integriruemykh gamiltonovykh sistem”, Matem. sb., 201:8 (2010), 63–102 | DOI | MR | Zbl

[13] Oshemkov A. A., “Sedlovye osobennosti slozhnosti 1 integriruemykh gamiltonovykh sistem”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2011, no. 2, 3–12

[14] Ryabov P. E., Kharlamov M. P., “Klassifikatsiya osobennostei v zadache o dvizhenii volchka Kovalevskoi v dvoinom pole sil”, Matem. sb., 203:2 (2012), 111–142 ; arXiv: 1408.0077 | DOI | MR | Zbl

[15] Celivanova E. N., “Topologiya zadachi o trëkhtochechnykh vikhryakh”, Tr. MIAN, 205, 1994, 141–149 | MR | Zbl

[16] Smeil C., “Topologiya i mekhanika”, UMN, 27:2(164) (1972), 77–133 | MR | Zbl

[17] Sokolov V. V., “Novyi integriruemyi sluchai dlya uravnenii Kirkhgofa”, Teor. i matem. fiz., 129:1 (2001), 31–37 | DOI | MR | Zbl

[18] Sokolov V. V., “Ob odnom klasse kvadratichnykh $\mathrm{so}(4)$-gamiltonianov”, Dokl. Akad. nauk, 394:5 (2004), 602–605 | MR | Zbl

[19] Sokolov V. V., Tsyganov A. V., “Pary Laksa dlya deformirovannykh volchkov Kovalevskoi i Goryacheva–Chaplygina”, Teor. i matem. fiz., 131:1 (2002), 118–125 | DOI | MR | Zbl

[20] Fomenko A. T., “Teoriya Morsa integriruemykh gamiltonovykh sistem”, DAN SSSR, 287:5 (1986), 1071–1075 | MR | Zbl

[21] Fomenko A. T., “Topologiya poverkhnostei postoyannoi energii nekotorykh integriruemykh gamiltonovykh sistem i prepyatstviya k integriruemosti”, Izv. AN SSSR. Ser. matem., 50:6 (1986), 1276–1307 | MR | Zbl

[22] Fomenko A. T., Simplekticheskaya geometriya. Metody i prilozheniya, Izd-vo Mosk. un-ta, M., 1988 | MR

[23] Fomenko A. T., “Simplekticheskaya topologiya vpolne integriruemykh gamiltonovykh sistem”, UMN, 44:1 (1989), 145–173 | MR | Zbl

[24] Fomenko A. T., “Teoriya bordizmov integriruemykh gamiltonovykh nevyrozhdennykh sistem s dvumya stepenyami svobody. Novyi topologicheskii invariant mnogomernykh integriruemykh sistem”, Izv. AN SSSR. Ser. matem., 55:4 (1991), 747–779 | MR | Zbl

[25] Fomenko A. T., “Topologicheskii invariant, grubo klassifitsiruyuschii integriruemye strogo nevyrozhdennye gamiltoniany na chetyrëkhmernykh simplekticheskikh mnogoobraziyakh”, Funkts. analiz i ego pril., 25:4 (1991), 23–35 | MR | Zbl

[26] Fomenko A. T., Tsishang X., “O tipichnykh topologicheskikh svoistvakh integriruemykh gamiltonovykh sistem”, Izv. AN SSSR. Ser. matem., 52:2 (1988), 378–407 | MR | Zbl

[27] Kharlamov M. P., “Integralnye mnogoobraziya privedënnoi sistemy v zadache o dvizhenii po inertsii tvërdogo tela s nepodvizhnoi tochkoi”, Mekh. tvërd. tela, 1976, no. 8, 18–23; arXiv: 1408.4548

[28] Kharlamov M. P., “Fazovaya topologiya odnogo integriruemogo sluchaya dvizheniya tvërdogo tela”, Mekh. tvërd. tela, 1979, no. 11, 50–64 ; arXiv: 1408.6028 | MR | Zbl

[29] Kharlamov M. P., “Bifurkatsii sovmestnykh urovnei pervykh integralov v sluchae Kovalevskoi”, Prikl. matem. i mekh., 47:6 (1983), 922–930 ; arXiv: 1312.7299 | MR | Zbl

[30] Kharlamov M. P., “Topologicheskii analiz klassicheskikh integriruemykh sistem v dinamike tvërdogo tela”, DAN SSSR, 273:6 (1983), 1322–1325 ; arXiv: 0906.2548 | MR | Zbl

[31] Kharlamov M. P., Topologicheskii analiz integriruemykh zadach dinamiki tvërdogo tela, Izd-vo LGU, L., 1988 ; arXiv: 1401.1015 | MR

[32] Kharlamov M. P., “Odin klass reshenii s dvumya invariantnymi sootnosheniyami zadachi o dvizhenii volchka Kovalevskoi v dvoinom postoyannom pole”, Mekh. tvërd. tela, 2002, no. 32, 32–38 ; arXiv: 0803.1028 | MR | Zbl

[33] Kharlamov M. P., “Kriticheskoe mnozhestvo i bifurkatsionnaya diagramma zadachi o dvizhenii volchka Kovalevskoi v dvoinom pole”, Mekh. tvërd. tela, 2004, no. 34, 47–58 | MR

[34] Kharlamov M. P., “Oblasti suschestvovaniya kriticheskikh dvizhenii obobschënnogo volchka Kovalevskoi i bifurkatsionnye diagrammy”, Mekh. tvërd. tela, 2006, no. 36, 13–22 | MR

[35] Kharlamov M. P., “Osobye periodicheskie resheniya obobschënnogo sluchaya Delone”, Mekh. tvërd. tela, 2006, no. 36, 23–33 | MR

[36] Kharlamov M. P., “Polnyi topologicheskii atlas integriruemoi sistemy s dvumya ili tremya stepenyami svobody”, Sovremennye metody teorii kraevykh zadach, Materialy Voronezhskoi vesennei matem. shkoly “Pontryaginskie chteniya – XXIV”, Izd-vo VGU, Voronezh, 2013

[37] Kharlamov M. P., Ryabov P. E., “Setevye diagrammy dlya invarianta Fomenko v integriruemoi sisteme s tremya stepenyami svobody”, Dokl. Akad. nauk, 447:5 (2012), 499–502 | MR | Zbl

[38] Kharlamov M. P., Savushkin A. Yu., “Razdelenie peremennykh i integralnye mnogoobraziya v odnoi chastnoi zadache o dvizhenii obobschënnogo volchka Kovalevskoi”, Ukr. matem. vestn., 1:4 (2004), 564–582 | MR

[39] Bogoyavlensky O. I., “Euler equations on finite-dimension Lie algebras arising in physical problems”, Commun. Math. Phys., 95 (1984), 307–315 | DOI | MR

[40] Bolsinov A. V., “Methods of calculation of the Fomenko–Zieschang invariant”, Adv. Sov. Math., 6, 1991, 147–183 | MR | Zbl

[41] Bolsinov A. V., Dullin H. R., Veselov A. P., “Spectra of Sol-manifolds: arithmetic and quantum monodromy”, Commun. Math. Phys., 264 (2006), 583–611 | DOI | MR | Zbl

[42] Bolsinov A. V., Oshemkov A. A., “Singularities of integrable Hamiltonian systems”, Topological Methods in the Theory of Integrable Systems, Cambridge Sci. Publ., Cambridge, 2006, 1–67 | MR | Zbl

[43] Brailov Yu. A., “Geometry of singularities of integrable systems on Lie algebras”, Rev. Math. Math. Phys., 12:2 (2005), 1–51 | Zbl

[44] Davison C. M., Dullin H. R., Bolsinov A. V., “Geodesics on the ellipsoid and monodromy”, J. Geom. Phys., 57 (2007), 2437–2454 | DOI | MR | Zbl

[45] Fomenko A. T., “The theory of invariants of multidimensional integrable Hamiltonian systems (with arbitrary many degrees of freedom). Molecular table of all integrable systems with two degrees of freedom”, Adv. Sov. Math., 6, 1991, 1–35 | MR | Zbl

[46] Kharlamov M. P., “Bifurcation diagrams of the Kowalevski top in two constant fields”, Regular Chaotic Dynam., 10:4 (2005), 381–398 ; arXiv: 0803.0893 | DOI | MR | Zbl

[47] Kharlamov M. P., “Separation of variables in the generalized 4th Appelrot class”, Regular Chaotic Dynam., 12:3 (2007), 267–280 ; arXiv: 0803.1024 | DOI | MR | Zbl

[48] Kharlamov M. P., “Separation of variables in the generalized 4th Appelrot class. II. Real solutions”, Regular Chaotic Dynam., 14:6 (2009), 621–634 | DOI | MR | Zbl

[49] Kharlamov M. P., “Complete topological atlases of some integrable systems with two and three degrees of freedom”, IV Int. Conf. “Geometry, Dynamics, Integrable Systems”, Book of Abstracts, RCD, Izhevsk, 2013

[50] Kharlamov M. P., “Phase topology of one system with separated variables and singularities of the symplectic structure”, J. Geom. Phys., 87 (2015), 248–265 ; arXiv: 1312.5184 | DOI | MR | Zbl

[51] Kharlamov M. P., Shvedov E. G., “On the existence of motions in the generalized 4th Appelrot class”, Regular Chaotic Dynam., 11:3 (2006), 337–342 | DOI | MR | Zbl

[52] Kharlamov M. P., Zotev D. B., “Nondegenerate energy surfaces of rigid body in two constant fields”, Regular Chaotic Dynam., 10:1 (2005), 15–19 | DOI | MR | Zbl

[53] Kharlamova I. I., Savushkin A. Y., “Bifurcation diagrams involving the linear integral of Yehia”, J. Phys. A Math. Theor., 43:10 (2010), 0301–0311 | DOI | MR

[54] Lerman L. M., Umanskiǐ Ya. L., “Structure of the Poisson action of $\mathbb R^2$ on a four-dimensional symplectic manifold. I”, Selecta Math. Sov., 6:4 (1987), 365–396 ; “II”, Selecta Math. Sov., 7:1 (1988), 39–48 | MR | Zbl | MR | Zbl

[55] Oshemkov A. A., “Fomenko invariants for the main integrable cases of the rigid body motion equations”, Adv. Sov. Math., 6, 1991, 67–146 | MR | Zbl

[56] Polyakova L. S., “Topological invariants for some algebraic analogs of the Toda lattice”, Adv. Sov. Math., 6, 1991, 185–207 | MR | Zbl

[57] Reyman A. G., Semenov-Tian-Shansky M. A., “Lax representation with a spectral parameter for the Kowalewski top and its generalizations”, Lett. Math. Phys., 14:1 (1987), 55–61 | DOI | MR | Zbl

[58] Sokolov V. V., “A generalized Kowalewski Hamiltonian and new integrable cases on $e(3)$ and $\mathrm{so}(4)$”, Kowalevski Property, CRM Proc. Lect. Notes, ed. V. B. Kuznetsov, Amer. Math. Soc., Providence, 2002, 304–315 | MR

[59] Yehia H. M., “New integrable cases in the dynamics of rigid bodies”, Mech. Research Commun., 13:3 (1986), 169–172 | DOI | MR | Zbl

[60] Yehia H. M., “New integrable problems in the dynamics of rigid bodies with the Kovalevskaya configuration. I. The case of axisymmetric forces”, Mech. Research Commun., 23:5 (1996), 423–427 | DOI | MR

[61] Yehia H. M., “New integrable problems in the dynamics of rigid bodies with the Kovalevskaya configuration. II. The case of asymmetric forces”, Mech. Research Commun., 23:5 (1996), 429–431 | DOI | MR

[62] Yehia H. M., “The master integrable two-dimensional system with a quartic second integral”, J. Phys. A Math. Gen., 39 (2006), 5807–5824 | DOI | MR | Zbl

[63] Zotev D. B., “Fomenko–Zieschang invariant in the Bogoyavlenskyi case”, Regular Chaotic Dynam., 5:4 (2000), 437–458 | DOI | MR

[64] Zung N. T., “Decomposition of nondegenerate singularities of integrable Hamiltonian systems”, Lett. Math. Phys., 33 (1995), 187–193 | DOI | MR | Zbl