Topology and stability of integrable systems

A. V. Bolsinov   ; A. V. Borisov   ; I. S. Mamaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 65 (2010) no. 2, pp. 259-318

Résumé

In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are illustrated by examples of several integrable problems related to the classical Euler–Poisson equations, the motion of a rigid body in a fluid, and the dynamics of gaseous expanding ellipsoids. These topological methods also enable one to find non-degenerate periodic solutions of integrable systems, which is especially topical in those cases where no general solution (for example, by separation of variables) is known. Bibliography: 82 titles.

Détail

Keywords: topology, stability, periodic trajectory, critical set, bifurcation diagram.
Mots-clés : bifurcation set

Bibliographie

[1] S. Smale, “Topology and mechanics. I”, Invent. Math., 10:4 (1970), 305–331 ; S. Smale, “Topology and mechanics. II. The planar $n$-body problem”, Invent. Math., 11:1 (1970), 45–64 | DOI | DOI | MR | MR | MR | Zbl | Zbl | Zbl

[2] V. V. Kozlov, “Topological obstacles to the integrability of natural mechanical systems”, Soviet Math. Dokl., 20:6 (1970), 1413–1415 | MR | Zbl

[3] V. V. Kozlov, “Integrability and non-integrability in Hamiltonian mechanics”, Russian Math. Surveys, 38:1 (1983), 1–76 | DOI | MR | Zbl

[4] V. V. Kozlov, Symmetries, topology and resonances in Hamiltonian mechanics, Ergeb. Math. Grenzgeb. (3), 31, Springer-Verlag, Berlin, 1996, 378 pp. | MR | MR | Zbl | Zbl

[5] A. T. Fomenko, Symplectic geometry, Adv. Stud. Contemp. Math., 5, Gordon and Breach, New York, 1988, 387 pp. | MR | MR | Zbl | Zbl

[6] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658 | DOI | MR | Zbl

[7] A. T. Fomenko, Kh. Tsishang, “Topologicheskii invariant i kriterii ekvivalentnosti integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody”, Izv. AN SSSR. Ser. matem., 54:3 (1990), 546–575 ; A. T. Fomenko, Kh. Tsishang, “Topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Math. USSR-Izv., 36:3 (1991), 567–596 | MR | Zbl | DOI | Zbl

[8] T. I. Pogosian, M. P. Kharlamov, “Bifurcation set and integral manifolds of the problem concerning the motion of a rigid body in a linear force field”, J. Appl. Math. Mech., 43:3 (1979), 452–462 | DOI | MR | Zbl

[9] T. I. Pogosyan, “Postroenie bifurkatsionnykh mnozhestv v odnoi zadache dinamiki tverdogo tela”, Mekhanika tverdogo tela, 12, Naukova dumka, Kiev, 1980, 9–16 | MR | Zbl

[10] T. I. Pogosyan, “Oblasti vozmozhnosti dvizheniya i integralnye mnogoobraziya v zadache Klebsha. Regulyarnyi sluchai”, Mekhanika tverdogo tela, 16, Naukova dumka, Kiev, 1984, 12–19 | MR | Zbl

[11] T. I. Pogosyan, “Oblasti vozmozhnosti dvizheniya v zadache Klebsha. Kriticheskii sluchai”, Mekhanika tverdogo tela, 15, Naukova dumka, Kiev, 1983, 3–23 | MR | Zbl

[12] T. I. Pogosyan, “Oblasti vozmozhnosti dvizheniya v zadache Klebsha”, Mekhanika tverdogo tela, 16, Naukova dumka, Kiev, 1984, 19–24 | MR | Zbl

[13] M. P. Kharlamov, Topologicheskii analiz integriruemykh zadach dinamiki tverdogo tela, Izd-vo LGU, L., 1988, 200 pp. | MR

[14] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall, Boca Raton, FL, 2004, 730 pp. | MR | MR | Zbl | Zbl

[15] A. A. Oshemkov, “Fomenko invariants for the main integrable cases of the rigid body motion equations”, Topological classification of integrable systems, Adv. Soviet Math., 6, Amer. Math. Soc., Providence, RI, 1991, 67–146 | MR | Zbl

[16] M. P. Harlamov, P. E. Ryabov, “The bifurcations of the first integrals in the case of Kowalewski–Yehia”, Regul. Chaotic Dyn., 2:2 (1997), 25–40 | DOI | MR | Zbl

[17] O. E. Orel, P. E. Ryabov, “Bifurcation sets in a problem on motion of rigid body in fluid and in the generalization of this problem”, Regul. Chaotic Dyn., 3:2 (1998), 82–91 | DOI | MR | Zbl

[18] P. V. Morozov, “The Liouville classification of integrable systems of the Clebsch case”, Sb. Math., 193:10 (2002), 1507–1533 | DOI | MR | Zbl

[19] P. V. Morozov, “Topology of Liouville foliations in the Steklov and the Sokolov integrable cases of Kirchhoff's equations”, Sb. Math., 195:3 (2004), 369–412 | DOI | MR | Zbl

[20] M. Yu. Ivochkin, “Topological analysis of the motion of an ellipsoid on a smooth plane”, Sb. Math., 199:6 (2008), 871–890 | DOI | MR | Zbl

[21] D. B. Zotev, “Phase topology of Appelrot class I of a Kowalewski top in a magnetic field”, J. Math. Sci. (N. Y.), 149:1 (2008), 922–946 | DOI | MR

[22] A. V. Bolsinov, P. H. Richter, A. T. Fomenko, “The method of loop molecules and the topology of the Kovalevskaya top”, Sb. Math., 191:2 (2000), 151–188 | DOI | MR | Zbl

[23] M. Radnović, V. Rom-Kedar, “Foliations of isonergy surfaces and singularities of curves”, Regul. Chaotic Dyn., 13:6 (2008), 645–668 | DOI | MR

[24] H. R. Dullin, M. Juhnke, P. H. Richter, “Action integrals and energy surfaces of the Kovalevskaya top”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4:6 (1994), 1535–1562 | DOI | MR | Zbl

[25] H. Waalkens, H. R. Dullin, P. H. Richter, “The problem of two fixed centers: bifurcations, actions, monodromy”, Phys. D, 196:3–4 (2004), 265–310 | DOI | MR | Zbl

[26] E. Sinitsyn, B. Zhilinskii, “Qualitative analysis of the classical and quantum Manakov top”, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 046, 23 pp., arXiv: math-ph/0703045 | DOI | MR | Zbl

[27] J. J. Duistermaat, “On global action-angle coordinates”, Comm. Pure Appl. Math., 33:6 (1980), 687–706 | DOI | MR | Zbl

[28] L. H. Eliasson, “Normal forms for Hamiltonian systems with Poisson commuting integrals – elliptic case”, Comment. Math. Helv., 65:1 (1990), 4–35 | DOI | MR | Zbl

[29] L. M. Lerman, Ya. L. Umanskii, “Klassifikatsiya chetyrekhmernykh integriruemykh gamiltonovykh sistem i puassonovskikh deistvii $\mathbb R^2$ v rasshirennykh okrestnostyakh prostykh osobykh tochek. I”, Matem. sb., 183:12 (1992), 141–176 ; L. M. Lerman, Ya. L. Umanskii, “Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of $\mathbb R^2$ in extended neighborhoods of simple singular points. I”, Russian Acad. Sci. Sb. Math., 77:2 (1994), 511–542 ; “II”, МатРμРј. СЃР±., 184:4 (1993), 105–138 ; “II”, Russian Acad. Sci. Sb. Math., 78:2 (1994), 479–506 ; “III. Р Рμализация”, МатРμРј. СЃР±., 186:10 (1995), 89–102 ; “III. Realization”, Sb. Math., 186:10 (1995), 1477–1491 | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI

[30] R. H. Cushman, L. M. Bates, Global aspects of classical integrable systems, Birkhäuser, Basel, 1997, 435 pp. | MR | Zbl

[31] T. Delzant, “Hamiltoniens périodiques et images covexes de l'application moment”, Bull. Soc. Math. France, 116:3 (1988), 315–339 | MR | Zbl

[32] N. T. Zung, “Symplectic topology of integrable Hamiltonian systems. I: Arnold–Liouville with singularities”, Compos. Math., 101:2 (1996), 179–215 | MR | Zbl

[33] N. T. Zung, “Symplectic topology of integrable hamiltonian systems. II: Topological classification”, Compos. Math., 138:2 (2003), 125–156 | DOI | MR | Zbl

[34] I. A. Taimanov, “Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds”, Math. USSR-Izv., 30:2 (1988), 403–409 | DOI | MR | Zbl

[35] I. A. Taimanov, “O topologicheskikh svoistvakh integriruemykh geodezicheskikh potokov”, Matem. zametki, 44:2 (1988), 283–284 | MR | Zbl

[36] S. V. Bolotin, “Nonintegrability of the problem of $n$ centers for $n>2$”, Mosc. Univ. Mech. Bull., 39:3 (1984), 65–68 | MR | Zbl

[37] S. V. Bolotin, “First integrals of systems with gyroscopic forces”, Mosc. Univ. Mech. Bull., 39:6 (1984), 20–28 | MR | Zbl

[38] A. V. Bolsinov, I. A. Taimanov, “Integrable geodesic flows with positive topological entropy”, Invent. Math., 140:3 (2000), 639–650 | DOI | MR | Zbl

[39] G. P. Paternain, “On the topology of manifolds with completely integrable geodesic flows”, Ergodic Theory Dynam. Systems, 12:1 (1992), 109–121 | DOI | MR | Zbl

[40] L. T. Butler, “Invariant fibrations of geodesic flows”, Topology, 44:4 (2005), 769–789 | DOI | MR | Zbl

[41] L. T. Butler, “The Maslov cocycle, smooth structures and real-analytic complete integrability”, Amer. J. Math., 131:5 (2009), 1311–1336, arXiv: 0708.3157v2 | DOI | MR | Zbl

[42] B. S. Bardin, “K zadache ob ustoichivosti mayatnikoobraznykh dvizhenii tverdogo tela v sluchae Goryacheva–Chaplygina”, Izv. RAN. MTT, 2007, no. 2, 14–21

[43] A. Z. Bryum, “Orbital stability analysis using first integrals”, J. Appl. Math. Mech., 53:6 (1989), 689–695 | DOI | MR | Zbl

[44] A. V. Karapetyan, “Invariant sets in the Goryachev–Chaplygin problem: existence, stability and branching”, J. Appl. Math. Mech., 70:2 (2006), 195–198 | DOI | MR | Zbl

[45] A. V. Karapetyan, “Invariant sets in the Clebsch–Tisserand problem: Existence and stability”, J. Appl. Math. Mech., 70:6 (2006), 859–864 | DOI | MR | Zbl

[46] A. P. Markeyev, “The stability of the plane motions of a rigid body in the Kovalevskaya case”, J. Appl. Math. Mech., 65:1 (2001), 47–54 | DOI | MR | Zbl

[47] A. P. Markeev, S. V. Medvedev, T. N. Chekhovskaya, “K zadache ob ustoichivosti mayatnikovykh dvizhenii tverdogo tela v sluchae Kovalevskoi”, Izv. RAN. MTT, 2003, no. 1, 3–9

[48] A. P. Markeyev, “The pendulum-like motions of a rigid body in the Goryachev–Chaplygin case”, J. Appl. Math. Mech., 68:2 (2004), 249–258 | DOI | MR | Zbl

[49] A. Yu. Moskvin, “Shar Chaplygina s girostatom: osobye resheniya”, Nelineinaya dinamika, 5:3 (2009), 345–356 http://nd.ics.org.ru/doc/r/pdf/1547/0

[50] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, “Mathematical aspects of classical and celestial mechanics. Dynamical systems, III”, Encyclopaedia Math. Sci., 3, Springer, Berlin, 1993, vii–xiv+129 pp. | MR | MR | Zbl | Zbl

[51] A. V. Borisov, I. S. Mamaev, Dinamika tverdogo tela. Gamiltonovy metody, integriruemost, khaos, 2-e izd., NITs “RKhD”, Izhevsk, 2005, 576 pp. | MR | Zbl

[52] B. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967, 472 pp. | MR | Zbl

[53] A. V. Borisov, I. S. Mamaev, “Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems”, Regul. Chaotic Dyn., 13:5 (2008), 443–490 | DOI | MR

[54] K. Ehlers, J. Koiller, J. Montgomery, P. Rios, “Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization”, The breadth of symplectic and Poisson geometry, Progr. Math., 232, eds. J. E. Marsden, T. S. Ratiu, Birkhäuser, Boston, MA, 2005, 75–120 | MR | Zbl

[55] A. V. Bolsinov, “Methods of calculation of the Fomenko–Zieschang invariant”, Topological classification of integrable systems, Adv. Soviet Math., 6, Amer. Math. Soc., Providence, RI, 1991, 147–183 | MR | Zbl

[56] E. Miranda, N. T. Zung, “Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian system”, Ann. Sci. École Norm. Sup. (4), 37:6 (2004), 819–839 | DOI | MR | Zbl

[57] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73 (1967), 747–817 | DOI | MR | MR | Zbl

[58] H. R. Dullin, A. V. Ivanov, “Vanishing twist in the Hamiltonian Hopf bifurcation”, Phys. D, 201:1–2 (2005), 27–44 | DOI | MR | Zbl

[59] V. V. Kalashnikov, “Typical integrable Hamiltonian systems on a four-dimensional symplectic manifold”, Izv. Math., 62:2 (1998), 261–285 | DOI | MR | Zbl

[60] S. A. Chaplygin, “Novoe chastnoe reshenie zadachi o vraschenii tyazhelogo tverdogo tela vokrug nepodvizhnoi tochki”, Trudy otd. fiz. nauk obsch-va lyubitelei estestvoznaniya, antropologii i etnografii, 12:1 (1904), 1–4; РЎРѕР±СЂ. СЃРѕС‡., С‚. 1, Р“Р�РўРўР›, Рњ., Р›., 1948, 125–132 | MR

[61] V. V. Kozlov, Metody kachestvennogo analiza v dinamike tverdogo tela, 2-e izd., Izd-vo RKhD, Izhevsk, 2000, 248 pp. | MR | Zbl

[62] A. V. Tsiganov, “Separation of variables in the Kovalevskaya–Goryachev–Chaplygin gyrostat”, Theoret. Math. Phys., 135:2 (2003), 651–658 | DOI | MR

[63] A. V. Borisov, A. G. Kholmskaya, I. S. Mamaev, “S. V. Kovalevskaya top and generalizations of integrable systems”, Regul. Chaotic Dyn., 6:1 (2001), 1–16 | DOI | MR | Zbl

[64] A. V. Borisov, I. S. Mamaev, “Euler–Poisson equations and integrable cases”, Regul. Chaotic Dyn., 6:3 (2001), 253–276 | DOI | MR | Zbl

[65] A. Chenciner, R. Montgomery, “A remarkable periodic solution of the three-body problem in the case of equal masses”, Ann. of Math. (2), 152:3 (2000), 881–901, arXiv: math/0011268 | DOI | MR | Zbl

[66] A. Chenciner, J. Gerver, R. Montgomery, C. Simo, “Simple choreographies of $N$ bodies: a preliminary study”, Geometry, mechanics, and dynamics, eds. P. Newton, P. Holmes, A. Weinstein, Springer, New York, 2002, 287–308 | DOI | MR | Zbl

[67] A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Absolute and relative choreographies in rigid body dynamics”, Regul. Chaotic Dyn., 13:3 (2008), 204–220 | DOI | MR

[68] Sistema Klebsha. Razdelenie peremennykh, yavnoe integrirovanie?, eds. A. V. Borisov, A. V. Tsyganov, NITs “RKhD”, M., Izhevsk, 2009, 287 pp.

[69] F. Kötter, “Ueber die Bewegung eines festen Körpers in einer Flüssigkeit. I, II”, J. Reine Angew. Math., 109 (1892), 51–81 ; 89–111 | Zbl | Zbl

[70] V. G. Marikhin, V. V. Sokolov, “O privedenii pary kvadratichnykh po impulsam gamiltonianov k kanonicheskoi forme i o veschestvennom chastichnom razdelenii peremennykh dlya volchka Klebsha”, Nelineinaya dinamika, 4:3 (2008), 313–322 http://nd.ics.org.ru/doc/r/pdf/1316/0

[71] V. A. Steklov, “Odin sluchai dvizheniya tyazhelogo tverdogo tela v zhidkosti”, Trudy otd. fiz. nauk obsch-va lyubitelei estestvoznaniya, antropologii i etnografii, 7 (1895), 1–40

[72] G. Kolossoff, “On some cases of motion of a solid in infinite liquid”, Amer. J. Math., 28:4 (1906), 367–376 | DOI | MR | Zbl

[73] B. Gaffet, “Expanding gas clouds of ellipsoidal shape: new exact solutions”, J. Fluid Mech., 325 (1996), 113–144 | DOI | MR | Zbl

[74] B. Gaffet, “Spinning gas clouds – without vorticity”, J. Phys. A, 33:21 (2000), 3929–3946 | DOI | MR | Zbl

[75] A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Multiparticle systems. The algebra of integrals and integrable cases”, Regul. Chaotic Dyn., 14:1 (2009), 18–41 | DOI | MR

[76] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, “Lax representation with a spectral parameter for the Kovalevskaya top and its generalizations”, Funct. Anal. Appl., 22:2 (1988), 158–160 | DOI | MR

[77] O. I. Bogoyavlenskii, “Integrable Euler equations on Lie algebras arising in problems of mathematical physics”, Math. USSR-Izv., 25:2 (1985), 207–257 | DOI | MR | Zbl

[78] O. I. Bogoyavlenskii, Oprokidyvayuschiesya solitony. Nelineinye integriruemye uravneniya, Nauka, M., 1991, 320 pp. | MR | Zbl

[79] V. M. Alekseev, “Obobschennaya prostranstvennaya zadacha dvukh nepodvizhnykh tsentrov. Klassifikatsiya dvizhenii”, Byull. In-ta teoret. astronomii, 10:4(117) (1965), 241–271 | MR

[80] A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Lie algebras in vortex dynamics and celestial mechanics – IV”, Regul. Chaotic Dyn., 4:1 (1999), 23–50 | DOI | MR | Zbl

[81] A. V. Borisov, I. S. Mamaev, S. M. Ramodanov, “Motion of a circular cylinder and $n$ point vortices in a perfect fluid”, Regul. Chaotic Dyn., 8:4 (2003), 449–462 | DOI | MR | Zbl

[82] A. V. Borisov, I. S. Mamaev, S. M. Ramodanov, “Dynamic interaction of point vortices and a two-dimensional cylinder”, J. Math. Phys., 48:6 (2007), 065403, 9 pp. | DOI | MR | Zbl