Journal of Mathematical Sciences

, Volume 180, Issue 4, pp 365–530 | Cite as

Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems

Article

Abstract

This paper presents results concerning the geometric invariant theory of completely integrable Hamiltonian systems and also the classification of integrable cases of low-dimensional and high-dimensional rigid body dynamics in a nonconservative force field. The latter problems are described by dynamical systems with variable dissipation. The first part of the work is the basis for the doctorial dissertation of V. V. Trofimov (1953–2003), which was already published in parts. However, in the present complete form, it has not appeared, and we decided to fill in this gap. The second part is a development of the results presented in the doctoral dissertation of M. V. Shamolin and has not appeared in the present variant. These two parts complement one another well, which initiated this work (its sketches already appeared in 1997).

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References

  1. 1.
    S. A. Agafonov, D. V. Georgievskii, and M. V. Shamolin, “Certain actual problems of geometry and mechanics. Abstract of sessions of workshop ‘Some Actual Problems of Geometry and Mechanics’,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 34 (2007).Google Scholar
  2. 2.
    R. R. Aidagulov and M. V. Shamolin, “A certain improvement of the Convey algorithm,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 53–55 (2005).Google Scholar
  3. 3.
    R. R. Aidagulov and M. V. Shamolin, “Archimedean uniform structures. Abstracts of sessions of workshop ‘Actual problems of Geometry and Mechanics’,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 46–51 (2007).Google Scholar
  4. 4.
    R. R. Aidagulov and M. V. Shamolin, “Manifolds of uniform structures. Abstracts of sessions of workshop ‘Actual problems of Geometry and Mechanics’,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 71–86 (2007).Google Scholar
  5. 5.
    A. A. Andronov, A Collection of Works [in Russian], USSR Academy of Sciences, Moscow (1956).Google Scholar
  6. 6.
    A. A. Andronov and E. A. Leontovich, “Some cases of dependence of limit cycles on a parameter,” Uch. Zap. GGU, No. 6 (1937).Google Scholar
  7. 7.
    A. A. Andronov and E. A. Leontovich, “To theory of variations of qualitative structure of partition of the plane into trajectories,” Dokl. Akad. Nauk SSSR, 21, No. 9 (1938).Google Scholar
  8. 8.
    A. A. Andronov and E. A. Leontovich, “On the birth of limit cycles from a nonrough focus or center and from a nonrough limit cycle,” Mat. Sb., 40, No. 2 (1956).Google Scholar
  9. 9.
    A. A. Andronov and E. A. Leontovich, “On the birth of limit cycles from a separatrix loop and from an equilibrium state of saddle-node type,” Mat. Sb., 48, No. 3 (1959).Google Scholar
  10. 10.
    A. A. Andronov and E. A. Leontovich, “Dynamical systems of the first nonroughness degree on the plane,” Mat. Sb., 68, No. 3 (1965).Google Scholar
  11. 11.
    A. A. Andronov and E. A. Leontovich, “Sufficient conditions for the first-degree nonroughness of a dynamical system on the plane,” Differ. Uravn., 6, No. 12 (1970).Google Scholar
  12. 12.
    A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative Theory of Second-Order Dynamical Systems [in Russian], Nauka, Moscow (1966).Google Scholar
  13. 13.
    A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Bifurcation Theory of Dynamical Systems on the Plane [in Russian], Nauka, Moscow (1967)Google Scholar
  14. 14.
    A. A. Andronov and L. S. Pontryagin, “Rough systems,” Dokl. Akad. Nauk SSSR, 14, No. 5, 247–250 (1937).Google Scholar
  15. 15.
    D. V. Anosov, “Geodesic flows on closed Riemannian manifolds of negative curvature,” Tr. Mat. Inst. Akad. Nauk SSSR, 90 (1967).Google Scholar
  16. 16.
    S. Kh. Aronson and V. Z. Grines, “Topological classification of flows on closed two-dimensional manifolds,” Usp. Mat. Nauk, 41, No. 1 (1986)Google Scholar
  17. 17.
    V. I. Arnol’d, “On characteristic class entering quantization conditions,” Funkts. Anal. Prilozh., 1, No. 1, 1–14 (1967).MATHCrossRefGoogle Scholar
  18. 18.
    V. I. Arnol’d, “Hamiltonian property of Euler equations of rigid body dynamics in an ideal fluid,” Usp. Mat. Nauk, 24, No. 3, 225–226 (1969).MATHGoogle Scholar
  19. 19.
    V. I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989).Google Scholar
  20. 20.
    V. I. Arnol’d and A. B. Givental’, “Symplectic geometry,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 4, 5–139 (1985).MathSciNetGoogle Scholar
  21. 21.
    V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, “Mathematical aspects of classical mechanics,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 3 (1985).Google Scholar
  22. 22.
    D. Arrowsmith and C. Place, Ordinary Differential Equations, Qualitative Theory with Applications [Russian translation], Mir, Moscow (1986).Google Scholar
  23. 23.
    G. F. Baggis, “Rough systems of two differential equations,” Usp. Mat. Nauk, 10, No. 4 (1955).Google Scholar
  24. 24.
    E. A. Barashin and V. A. Tabueva, Dynamical Systems with Cylindrical Phase Space [in Russian], Nauka, Moscow (1969).Google Scholar
  25. 25.
    N. N. Bautin, “On number of limit cycles born under variation of coefficients from an equilibrium state of focus or center type,” Mat. Sb., 30, No. 1 (1952).Google Scholar
  26. 26.
    N. N. Bautin, “Some methods for qualitative studying dynamical systems related to field turn,” Prikl. Mat. Mekh., 37, No. 6 (1973).Google Scholar
  27. 27.
    N. N. Bautin and E. A. Leontovich, Methods and Tools for Qualitative Studying Dynamical Systems on Plane [in Russian], Nauka, Moscow (1976).Google Scholar
  28. 28.
    I. Bendixon, “On curves defined by differential equations,” Usp. Mat. Nauk, 9 (1941).Google Scholar
  29. 29.
    M. Berger, Geometry [Russian translation], Mir, Moscow (1984).Google Scholar
  30. 30.
    A. L. Besse, Manifolds All of Whose Geodesics Are Closed [Russian translation], Mir, Moscow (1981).Google Scholar
  31. 31.
    J. Birkhoff, Dynamical Systems [Russian translation], Gostekhizdat, Moscow (1941).Google Scholar
  32. 32.
    J. A. Bliess, Lectures on the Calculus of Variations [Russian translation], Gostekhizdat, Moscow–Leningrad (1941).Google Scholar
  33. 33.
    O. I. Bogoyavlenskii, “Dynamics of a rigid body with n ellipsoidal holes filled with a magnetic fluid,” Dokl. Akad. Nauk SSSR, 272, No. 6, 1364–1367 (1983).MathSciNetGoogle Scholar
  34. 34.
    O. I. Bogoyavlenskii, “Some integrable cases of Euler equations,” Dokl. Akad. Nauk SSSR, 287, No. 5, 1105–1108 (1986).MathSciNetGoogle Scholar
  35. 35.
    O. I. Bogoyavlenskii and G. F. Ivakh, “Topological analysis of V. A. Steklov integrable cases,” Usp. Mat. Nauk, 40, No. 4, 145–146 (1985).MathSciNetGoogle Scholar
  36. 36.
    A. V. Bolsinov, “Completeness criterion of a family of functions in involution constructed by argument shift method,” Dokl. Akad. Nauk SSSR, 301, No. 5, 1037–1040 (1988).Google Scholar
  37. 37.
    N. Bourbaki, Integration [Russian translation], Nauka, Moscow (1970).Google Scholar
  38. 38.
    N. Bourbaki, Lie Groups and Algebras [Russian translation], Mir, Moscow (1972).Google Scholar
  39. 39.
    A. V. Brailov, “Involutive tuples on Lie algebras and scalar field extension,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 7–51 (1983).Google Scholar
  40. 40.
    A. V. Brailov, “Some cases of complete integrability of Euler equations and applications,” Dokl. Akad. Nauk SSSR, 268, No. 5, 1043–1046 (1983).MathSciNetGoogle Scholar
  41. 41.
    A. D. Bryuno, Local Method for Nonlinear Analysis of Differential Equations [in Russian], Nauka, Moscow (1979).Google Scholar
  42. 42.
    V. S. Buslaev and V. A. Nalimova, “Trace formula in Lagrangian Mechanics,” Teor. Mat. Fiz., 61, No. 1, 52–63 (1984).MathSciNetGoogle Scholar
  43. 43.
    N. N. Butenina, “Bifurcations of separatrices and limit cycles of a two-dimensional dynamical system under field turn,” Differ. Uravn., 9, No. 8 (1973).Google Scholar
  44. 44.
    N. N. Butenina, “To bifurcation theory of dynamical systems under field turn,” Differ. Uravn., 10, No. 7 (1974).Google Scholar
  45. 45.
    M. L. Byalyi, “On first polynomial integrals in impulses for a mechanical system on two-dimensional torus,” Funkts. Anal. Prilozh., 21, No. 4, 64–65 (1987).MathSciNetGoogle Scholar
  46. 46.
    S. A. Chaplygin, Selected Works, [in Russian], Nauka, Moscow (1976).Google Scholar
  47. 47.
    Dao Chiog Tkhi and A. T. Fomenko, Minimal Surfaces and Plateau Problem [in Russian], Nauka, Moscow (1987).Google Scholar
  48. 48.
    P. Dazord, “Invariants homotopiques attachés aux fibrés simplectiques,” Ann. Inst. Fourier (Grenoble), 29, No. 2, 25–78 (1979).MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    R. Delanghe, F. Sommen, and V. Sousek, Clifford Algebra and Spinor-Valued Functions, Kluwer Academic, Dordrecht (1992).MATHCrossRefGoogle Scholar
  50. 50.
    F. M. Dimentberg, Theory of Spatial Hinge Mechanisms [in Russian], Nauka, Moscow (1982).Google Scholar
  51. 51.
    J. Dixmier, Universal Enveloping Algebras [Russian translation], Mir, Moscow (1978).Google Scholar
  52. 52.
    C. T. G. Dodson, Categories, Bundles and Spacetime Topology, Kluwer Academic, Dordrecht (1988).MATHGoogle Scholar
  53. 53.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable system. I,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 4, 179–284 (1985).MathSciNetGoogle Scholar
  54. 54.
    B. A. Dubrovin and S. P. Novikov, “On Poisson brackets of hydrodynamic type,” Dokl. Akad. Nauk SSSR, 279, No. 2, 294–297 (1984).MathSciNetGoogle Scholar
  55. 55.
    B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry, Methods and Applications [in Russian], Nauka, Moscow (1986).Google Scholar
  56. 56.
    A. T. Fomenko, Symplectic Geometry [in Russian], Izd. Mosk. Unuv., Moscow (1988).Google Scholar
  57. 57.
    A. T. Fomenko, “Bordism theory of integrable Hamiltonian nondegenerate systems with two degrees of freedom. A new topological invariant of many-dimensional Hamiltonian systems,” Izv. Akad. Nauk SSSR, Ser. Mat., 55, No. 4, 747–779 (1991).MATHGoogle Scholar
  58. 58.
    A. T. Fomenko and D. B. Fuks, A Course of Homotopical Topology [in Russian], Nauka, Moscow (1989).Google Scholar
  59. 59.
    A. T. Fomenko and V. V. Trofimov, Integrable Systems on Lie Algebras and Symmetric Spaces, Gordon and Breach, New York (1988).Google Scholar
  60. 60.
    H. Fujimoto, “Examples of complete minimal surfaces in \( {\mathbb{R}^m} \) whose Gauss maps omit \( \frac{{m\left( {m + 1} \right)}}{2} \) hyperplanes in general positions,” Sci. Rep. Kanazawa Univ., 33, No. 2, 37–43 (1989).MATHMathSciNetGoogle Scholar
  61. 61.
    D. B. Fuks, “On Maslov–Arnol’d characteristic classes,” Dokl. Akad. Nauk SSSR, 178, No. 2, 303–306 (1968).MathSciNetGoogle Scholar
  62. 62.
    D. V. Georgievskii and M. V. Shamolin, “Kinematics and geometry of masses of a rigid body with a fixed point in \( {\mathbb{R}^n} \),” Dokl. Ross. Akad. Nauk, 380, No. 1, 47–50 (2001).MathSciNetGoogle Scholar
  63. 63.
    D. V. Georgievskii and M. V. Shamolin, “On kinematics of a rigid body with a fixed point in \( {\mathbb{R}^n} \). Abstracts of sessions of workshop ‘Actual Problems of Geometry and Mechanics’,” Fundam. Prikl. Mat., 7, No. 1, 315 (2001).Google Scholar
  64. 64.
    D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with a fixed point in \( {\mathbb{R}^n} \),” Dokl. Ross. Akad. Nauk, 383, No. 5, 635–637 (2002).MathSciNetGoogle Scholar
  65. 65.
    D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion of generalized gyroscope in \( {\mathbb{R}^n} \),” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 37–41 (2003).Google Scholar
  66. 66.
    D. V. Georgievskii and M. V. Shamolin, “Valerii Vladimirovich Trofimov,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 5–6 (2007).Google Scholar
  67. 67.
    D. V. Georgievskii and M. V. Shamolin, “On kinematics of a rigid body with a fixed point in \( {\mathbb{R}^n} \). Abstracts of sessions of workshop ‘Actual Problems of Geometry and Mechanics’,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 24–25 (2007).Google Scholar
  68. 68.
    D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with a fixed point in \( {\mathbb{R}^n} \). Abstracts of sessions of workshop ‘Actual Problems of Geometry and Mechanics’,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 30 (2007).Google Scholar
  69. 69.
    D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion of generalized gyroscope in n-dimensional space. Abstracts of sessions of workshop ‘Actual Problems of Geometry and Mechanics’,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 31 (2007).Google Scholar
  70. 70.
    D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “Geometry and mechanics: problems, approaches, and methods. Abstracts of sessions of workshop ‘Actual Problems of Geometry and Mechanics’,” Fundam. Prikl. Mat., 7, No. 1, 301 (2001).Google Scholar
  71. 71.
    D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “On some topological invariants of flows with complex potential. Abstracts of sessions of workshop ‘Actual Problems of Geometry and Mechanics’,” Fundam. Prikl. Mat., 7, No. 1, 305 (2001).Google Scholar
  72. 72.
    D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “Geometry and mechanics; problems, approaches, and methods. Abstracts of sessions of workshop ‘Actual Problems of Geometry and Mechanics’,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 16 (2007).Google Scholar
  73. 73.
    D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “On some topological invariants of flows with complex potential. Abstracts of sessions of workshop ‘Actual Problems of Geometry and Mechanics’,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 19 (2007).Google Scholar
  74. 74.
    V. V. Golubev, Lectures on Analytic Theory of Differential Equations [in Russian], Gostekhizdat, Moscow–Leningrad (1950).Google Scholar
  75. 75.
    V. V. Golubev, Lectures on Integrating Equations of Motion of a Heavy Rigid Body Around a Fixed Point [in Russian], Gostekhizdat, Moscow–Leningrad (1953).Google Scholar
  76. 76.
    G. V. Gorr, L. V. Kudryashova, and L. A. Stepanova, Classical Problems of Rigid Body Dynamics [in Russian], Naukova Dumka, Kiev (1978).Google Scholar
  77. 77.
    D. N. Goryachev, “New cases of integrability of dynamical Euler equations,” Varshav. Univ. Izvestiya, Book 3, 1–15 (1916).Google Scholar
  78. 78.
    M. De Gosson, “La définition de l’indice de Maslov sans hypothèse de transversalité,” C. R. Acad. Sci. Paris, 310, 279–282 (1990).MATHGoogle Scholar
  79. 79.
    M. Goto and F. Grosshans, Semisimple Lie Algebras [Russian translation], Mir, Moscow (1981).Google Scholar
  80. 80.
    P. Griffiths, Exterior Differential forms and Calculus of Variations [Russian translation], Mir, Moscow (1986).Google Scholar
  81. 81.
    D. M. Grobman, “On homeomorphism of systems of differential equations,” Dokl. Akad. Nauk SSSR, 128, No. 5, 880–881 (1959).MATHMathSciNetGoogle Scholar
  82. 82.
    D. M. Grobmam, “Topological classification of neighborhoods of a singular point in n-dimensional space,” Mat. Sb., 56, No. 1, 77–94 (1962).MathSciNetGoogle Scholar
  83. 83.
    M. Gromov, “Pseudo-holomorphic curves in symplectic manifolds,” Invent. Math., 82, 307–347 (1985).MATHMathSciNetCrossRefGoogle Scholar
  84. 84.
    V. Guillemin and S. Sternberg, Geometric Asymptotics [Russian translation], Mir, Moscow (1981).Google Scholar
  85. 85.
    V. Guillemin and S. Sternberg, Symplectic Technique in Physics, Cambridge Univ. Press, Cambridge (1984).Google Scholar
  86. 86.
    Ph. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).Google Scholar
  87. 87.
    R. Harvey and H. B. Lawson, “Calibrated geometry,” Acta Math., 148, 47–157 (1982).MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    S. Helgason, Differential Geometry and Symmetric Spaces [Russian translation], Mir, Moscow (1964).Google Scholar
  89. 89.
    H. Hess, “Connections on symplectic manifolds and geometric quantizations,” in: Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lect. Notes Math., Vol. 836, Springer, Berlin (1980), pp. 153–166.Google Scholar
  90. 90.
    M. D. Hvidsten, “Volume and energy stability for isometric minimal immersions,” Illinois J. Math., 33, No. 3, 488–494 (1989).MATHMathSciNetGoogle Scholar
  91. 91.
    M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets, Geometry and Quantization [in Russian], Nauka, Moscow (1991).Google Scholar
  92. 92.
    M. Karasev and Yu. Vorobijev, “On analog of Maslov class in non-Lagrangian case,” in: Problems of Geometry, Topology and Mathematical Physics, New Developments in Global Analysis ser., Voronezh Univ. Press, Voronezh (1992), pp. 37–48.Google Scholar
  93. 93.
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 1, Nauka, Moscow (1981).Google Scholar
  94. 94.
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 2, Nauka, Moscow (1981).Google Scholar
  95. 95.
    A. N. Kolmogorov, “General theory of dynamical systems and classical mechanics,” in: Int. Math. Congress in Amsterdam [in Russian], Fizmatgiz, Moscow (1961), pp. 187–208.Google Scholar
  96. 96.
    A. I. Kostrikin, An Introduction to Algebra [in Russian], Nauka, Moscow (1977).Google Scholar
  97. 97.
    V. V. Kozlov, Methods of Qualitative Analysis in Rigid Body Dynamics [in Russian], Izd. Mosk. Univ., Moscow (1980).Google Scholar
  98. 98.
    V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,” Usp. Mat. Nauk, 38, No. 1, 3–67 (1983).Google Scholar
  99. 99.
    V. V. Kozlov and N. N. Kolesnikov, “On integrability of Hamiltonian systems,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 6, 88–91 (1979).Google Scholar
  100. 100.
    L. D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media [in Russian], Nauka, Moscow (1982).Google Scholar
  101. 101.
    S. Lang, Introduction to Theory of Differentiable Manifolds [Russian translation], Mir, Moscow (1967).Google Scholar
  102. 102.
    Le Hgok T’euen, “Complete involutive tuples of functions on extensions of Lie algebras related to Frobenius algebras,” Tr. Semin. Vekt. Tenz. Anal., No. 22, 69–106 (1985).Google Scholar
  103. 103.
    Le Hong Van, “Calibrations, minimal surfaces, and Maslov–Trofimov index,” in: Selected Problems in Algebra, Geometry, and Discrete Mathematics [in Russian], Moscow (1988), pp. 62–79.Google Scholar
  104. 104.
    Le Hong Van and A. T. Fomenko, “Minimality criterion of Lagrangian submanifolds in Kälerian manifolds,” Mat. Zametki, 46, No. 4, 559–571 (1987).Google Scholar
  105. 105.
    S. Leftschets, Geometric Theory of Differential Equations [Russian translation], IL, Moscow (1961).Google Scholar
  106. 106.
    V. G. Lemlein, “On spaces of symmetric and almost symmetric connection,” Dokl. Akad. Nauk SSSR, 116, No. 4, 655–658 (1957).MathSciNetGoogle Scholar
  107. 107.
    V. G. Lemlein, “Curvature tensor and certain types of spaces of symmetric and almost symmetric connection,” Dokl. Akad. Nauk SSSR, 117, No. 5, 755–758 (1957).MathSciNetGoogle Scholar
  108. 108.
    E. A. Leontovich and A. G. Maier, “On trajectories defining the qualitative structure of partition of a sphere into trajectories,” Dokl. Akad. Nauk SSSR, 14, No. 5, 251–254 (1937).Google Scholar
  109. 109.
    E. A. Leontovich and A. G. Maier, “On scheme defining topological structure of partition into trajectories,” Dokl. Akad. Nauk SSSR, 103, No. 4 (1955).Google Scholar
  110. 110.
    Yu. I. Levi, “On affine connections adjacent to a skew-symmetric tensor,” Dokl. Akad. Nauk SSSR, 128, No. 4, 668–671 (1959).MathSciNetGoogle Scholar
  111. 111.
    A. Lichnerowicz, “Deformation of algebras associated with a symplectic manifold,” in: M. Cahen et al., eds., Differential Geometry and Mathematical Physics (Liège, 1980; Leuven, 1981), Reidel, Dordrecht (1983), pp. 69–83.Google Scholar
  112. 112.
    J. Lion and M. Vergne, Weyl Representation, Maslov Index, and Theta-Series [Russian translation], Mir, Moscow (1983).Google Scholar
  113. 113.
    A. M. Lyapunov, “A new case of integrability of equations of motion of a rigid body in a fluid,” in: A Collection of Works [in Russian], Vol. 1, Akad. Nauk SSSR, Moscow (1954), pp. 320–324.Google Scholar
  114. 114.
    Yu. I. Manin, “Algebraic aspects of nonlinear differential equations,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 11, 5–112 (1978).MathSciNetGoogle Scholar
  115. 115.
    O. V. Manturov, “Homogeneous spaces and invariant tensors,” Itogi Nauki Tekh. Ser. Probl. Geom., 18, 105–142 (1986).MathSciNetGoogle Scholar
  116. 116.
    J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York (1976).MATHCrossRefGoogle Scholar
  117. 117.
    J. Marsden and A. Weinstein, “Reduction of symplectic manifolds with symmetry,” Rep. Math. Phys., 5, No. 1, 121–130 (1974).MATHMathSciNetCrossRefGoogle Scholar
  118. 118.
    V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Izd. Mosk. Univ., Moscow (1965).Google Scholar
  119. 119.
    V. P. Maslov, Asymptotic Methods and Perturbation Theory [in Russian], Nauka, Moscow (1988).Google Scholar
  120. 120.
    W. Miller, Symmetry and Separation of Variables [Russian translation], Mir, Moscow (1981).Google Scholar
  121. 121.
    R. Miron and V. Oproiu, “Almost cosymplectic and conformal almost cosymplectic connections,” Revue Roum. Math. Pures Appl., 16, No. 6, 893–912 (1971).MATHMathSciNetGoogle Scholar
  122. 122.
    A. S. Mishchenko, B. Yu. Sternin, and V. E. Shatalov, Lagrangian Manifolds and Canonical Operator Method [in Russian], Nauka, Moscow (1978).Google Scholar
  123. 123.
    A. S. Mishchenko and A. T. Fomenko, “Euler equations on finite-dimensional Lie groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 42, No. 2, 386–415 (1978).Google Scholar
  124. 124.
    J. M. Morvan, “Classes de Maslov d’une immersion Lagrangienne et minimalit´e,” C. R. Acad. Sci. Paris, 292, 633–636 (1981).MATHMathSciNetGoogle Scholar
  125. 125.
    Yu. I. Neimark, “On motions close to double-asymptotic motion,” Dokl. Akad. Nauk SSSR, 172, No. 5, 1021–1024 (1957).MathSciNetGoogle Scholar
  126. 126.
    Yu. I. Neimark, “Structure of motions of a dynamical system near a neighborhood of a homoclinic curve,” in: 5th Summer Mathematical School [in Russian], Kiev (1968), pp. 400–435.Google Scholar
  127. 127.
    V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow–Leningrad (1949).Google Scholar
  128. 128.
    Z. Nitetski, Introduction to Differential Dynamics [Russian translation], Mir, Moscow (1975).Google Scholar
  129. 129.
    E. Novak, “Generalized Maslov–Trofimov index for certain families of Hamiltonians on Lie algebras of upper-triangular matrices,” in: Algebra, Geometry, and Discrete Mathematics in Discrete Problems [in Russian], Moscow (1991), pp. 112–135.Google Scholar
  130. 130.
    S. P. Novikov, “Hamiltonian formalism and a multivalued analog of Morse theory,” Usp. Mat. Nauk, 37, No. 5, 3–49 (1982).Google Scholar
  131. 131.
    S. P. Novikov and I. Shmel’tser, “Periodic solutions of Kirchhoff equation of free motion of a rigid body and ideal fluid and extended Lyusternik–Shnirel’man–Morse theory, Funkts. Anal. Prilozh., 15, No. 3, 54–66 (1991).MathSciNetGoogle Scholar
  132. 132.
    J. Palais and S. Smale, “Structural stability theorems,” in: Mathematics (collection of translations) [in Russian], 13, No. 2, 145–155 (1969).Google Scholar
  133. 133.
    J. Palis and W. De Melu, Geometric Theory of Dynamical Systems. An Introduction [Russian translation], Mir, Moscow (1986).Google Scholar
  134. 134.
    J. Patera, R. T. Sharp, and P. Winternitz, “Invariants of real low dimension Lie algebras,” J. Math. Phys., 17, No. 6, 986–994 (1976).MATHMathSciNetCrossRefGoogle Scholar
  135. 135.
    M. Peixoto, “On structural stability,” Ann. Math. (2), 69, 199–222 (1959).MATHMathSciNetCrossRefGoogle Scholar
  136. 136.
    M. Peixoto, “Structural stability on two-dimensional manifolds,” Topology, 1, No. 2, 101–120 (1962).MATHMathSciNetCrossRefGoogle Scholar
  137. 137.
    M. Peixoto, “On an approximation theorem of Kupka and Smale,” J. Differ. Equ., 3, 214–227 (1966).MathSciNetCrossRefGoogle Scholar
  138. 138.
    V. A. Pliss, “On roughness of differential equations given on torus,” Vestn. Leningr. Univ. Mat., 13, 15–23 (1960).MathSciNetGoogle Scholar
  139. 139.
    V. A. Pliss, Integral Sets of Periodic Systems of Differential Equations [in Russian], Nauka, Moscow (1967).Google Scholar
  140. 140.
    H. Poincaré, On Curves Defined by Differential Equations [Russian translation], OGIZ, Moscow–Leningrad (1947).Google Scholar
  141. 141.
    H. Poincaré, “New methods in celestial mechanics,” in: Selected Works [Russian translation], Vols. 1. 2, Nauka, Moscow (1971, 1972).Google Scholar
  142. 142.
    H. Poincaré, On Science [Russian translation], Nauka, Moscow (1983).Google Scholar
  143. 143.
    A. G. Reiman, “Integrable Hamiltonian systems related to graded Lie algebras,” Zap. Nauchn. Sem. LOMI Akad. Nauk SSSR, 95, 3–54 (1980).MathSciNetGoogle Scholar
  144. 144.
    B. L. Reinhart, “Cobordism and Euler number,” Topology, 2, 173–178 (1963).MATHMathSciNetCrossRefGoogle Scholar
  145. 145.
    S. T. Sadetov, “Integrability conditions for Kirchhoff equations,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 56–62 (1990).Google Scholar
  146. 146.
    S. Salamon, Riemannian Geometry and Holonomy Group Research Notes Math. Ser., Vol. 201, Wiley, New York (1989).Google Scholar
  147. 147.
    E. J. Saletan, “Contraction of Lie groups,” J. Math. Phys., 2, No. 1, 1–22 (1961).MATHMathSciNetCrossRefGoogle Scholar
  148. 148.
    T. V. Sal’nikova, “On integrability of Kirchhoff equations in symmetric case,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 4, 68–71 (1985).Google Scholar
  149. 149.
    V. A. Samsonov and M. V. Shamolin, “To the problem of body motion in a resisting medium,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 51–54 (1989).Google Scholar
  150. 150.
    R. Schoen and S. T. Yau, “Complete three dimensional manifolds with positive Ricci curvature and scalar curvature,” Ann. Math. Stud., 209–228 (1982).Google Scholar
  151. 151.
    H. Seifert and W. Threlfall, Topology [Russian translation], Gostekhizdat, Moscow (1938).Google Scholar
  152. 152.
    J. P. Serre, “Singular homologies of fiber spaces,” in: Fiber Spaces and Applications [Russian translation], IL, Moscow (1958), pp. 9–114.Google Scholar
  153. 153.
    M. V. Shamolin, “Closed trajectories of different topological types in problem of body motion in a medium with resistance,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 52–56 (1992).Google Scholar
  154. 154.
    M. V. Shamolin, “To problem on body motion in a medium with resistance,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 52–58 (1992).Google Scholar
  155. 155.
    M. V. Shamolin, “Application of Poincaré topographical system and comparison system methods in some concrete systems of differential equations,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 66–70 (1993).Google Scholar
  156. 156.
    M. V. Shamolin, “Classification of phase portraits in problem of body motion in a resisting medium under existence of a linear damping moment,” Prikl. Mat. Mekh., 57, No. 4, 40–49 (1993).MathSciNetGoogle Scholar
  157. 157.
    M. V. Shamolin, “Existence and uniqueness of trajectories having infinitely distant points as limit sets for dynamical systems on the plane,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 68–71 (1993).Google Scholar
  158. 158.
    M. V. Shamolin, “A new two-parameter family of phase portraits in problem of body motion in a medium,” Dokl. Ross. Akad. Nauk, 337, No. 5, 611–614 (1994).MathSciNetGoogle Scholar
  159. 159.
    M. V. Shamolin, “On relative roughness of dynamical systems in problem of body motion in a resisting medium,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 6, 17 (1995).Google Scholar
  160. 160.
    M. V. Shamolin, “Definition of relative roughness and a two-dimensional phase portrait family in rigid body dynamics,” Usp. Mat. Nauk, 51, No. 1, 175–176 (1996).MathSciNetGoogle Scholar
  161. 161.
    M. V. Shamolin, “Introduction to problem of body drag in a resisting medium and a new two-parameter family of phase portraits,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 4, 57–69 (1996).Google Scholar
  162. 162.
    M. V. Shamolin, “Variety of phase portrait types in dynamics of a rigid body interacting with a resisting medium,” Dokl. Ross. Akad. Nauk, 349, No. 2, 193–197 (1996).MathSciNetGoogle Scholar
  163. 163.
    M. V. Shamolin, “On integrable case in spatial dynamics of a rigid body interacting with a medium,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 65–68 (1997).Google Scholar
  164. 164.
    M. V. Shamolin, “Spatial Poincaré topographical systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997).MathSciNetGoogle Scholar
  165. 165.
    M. V. Shamolin, “A family of portraits with limit cycles in plane dynamics of a rigid body interacting with a medium,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 6, 29–37 (1998).Google Scholar
  166. 166.
    M. V. Shamolin, “On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998).MathSciNetGoogle Scholar
  167. 167.
    M. V. Shamolin, “Some classical problems in a three-dimensional dynamics of a rigid body interacting with a medium,” in: Proc. of ICTACEM’98, Kharagpur, India, December 1–5, 1998, CD: Aerospace Engineering Dep., Indian Inst. of Technology, Kharagpur, India (1998).Google Scholar
  168. 168.
    M. V. Shamolin, “New Jacobi integrable cases in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999).Google Scholar
  169. 169.
    M. V. Shamolin, “On roughness of dissipative systems and relative roughness and nonroughness of variable dissipation systems,” Usp. Mat. Nauk, 54, No. 5, 181–182 (1999).MathSciNetGoogle Scholar
  170. 170.
    M. V. Shamolin, “Some classes of particular solutions in dynamics of a rigid body interacting with a medium,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 178–189 (1999).Google Scholar
  171. 171.
    M. V. Shamolin, “Structural stability in 3D dynamics of a rigid,” in: Proc. of WCSMO-3, Buffalo, NY, May 17–21, 1999, CD: Buffalo, NY (1999).Google Scholar
  172. 172.
    M. V. Shamolin, “A new family of phase portraits in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 371, No. 4, 480–483 (2000).MathSciNetGoogle Scholar
  173. 173.
    M. V. Shamolin, “Jacobi integrability in problem of motion of a four-dimensional rigid body in a resisting medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000).Google Scholar
  174. 174.
    M. V. Shamolin, “New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium,” in: Abstracts of 16th IMACS World Congress 2000, Lausanne, Switzerland, August 21–25, 2000, CD: EPFL (2000).Google Scholar
  175. 175.
    M. V. Shamolin, “On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000).MathSciNetGoogle Scholar
  176. 176.
    M. V. Shamolin, “On roughness of dissipative systems and relative roughness of variable dissipation systems,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 63 (2000).Google Scholar
  177. 177.
    M. V. Shamolin, “Complete integrability of equations of motion of a spatial pendulum in an over-run medium flow,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 22–28 (2001).Google Scholar
  178. 178.
    M. V. Shamolin, “Integrability cases of equations of spatial rigid body dynamics,” Prikl. Mekh., 37, No. 6, 74–82 (2001).MATHMathSciNetGoogle Scholar
  179. 179.
    M. V. Shamolin, “On integration of some classes of nonconservative systems, Usp. Mat. Nauk, 57, No. 1, 169–170 (2002).MathSciNetGoogle Scholar
  180. 180.
    M. V. Shamolin, “Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium,” J. Math. Sci., 110, No. 2, 2526–2555 (2002).MathSciNetCrossRefGoogle Scholar
  181. 181.
    M. V. Shamolin, “Foundations of differential and topological diagnostics,” J. Math. Sci., 114, No. 1, 976–1024 (2003).MATHMathSciNetCrossRefGoogle Scholar
  182. 182.
    M. V. Shamolin, “New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium,” J. Math. Sci., 114, No. 1, 919–975 (2003).MATHMathSciNetCrossRefGoogle Scholar
  183. 183.
    M. V. Shamolin, “Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body,” J. Math. Sci., 122, No. 1, 2841–2915 (2004).MATHMathSciNetCrossRefGoogle Scholar
  184. 184.
    M. V. Shamolin, “Geometric representation of motion in a certain problem of body interaction with a medium,” Prikl. Mekh., 40, No. 2, 137–14 (2004).MATHMathSciNetGoogle Scholar
  185. 185.
    M. V. Shamolin, “A case of complete integrability in spatial dynamics of a rigid body interacting with a medium taking account of rotational derivatives of force moment in angular velocity,” Dokl. Ross. Akad. Nauk, 403, No. 4. 482–485 (2005).MathSciNetGoogle Scholar
  186. 186.
    M. V. Shamolin, “Comparison of Jacobi integrability cases of plane and spatial body motion in a medium under streamline flow around,” Prikl. Mat. Mekh., 69, No. 6, 1003–1010 (2005).MATHMathSciNetGoogle Scholar
  187. 187.
    M. V. Shamolin, “On a certain integrability case of equations of dynamics on \( {\text{so}}(4) \times {\mathbb{R}^n} \),” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005).MathSciNetGoogle Scholar
  188. 188.
    M. V. Shamolin, “Structural stable vector fields in rigid body dynamics,” Proc. 8th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2005), Lodz, Poland, December 12–15, 2005, Vol. 1, Tech. Univ. Lodz (2005), pp. 429–436.Google Scholar
  189. 189.
    M. V. Shamolin, “Cases of complete integrability in elementary functions of some classes of nonconservative dynamical systems,” in: Abstracts of Reports of Int. Conf. “Classical Problems of Rigid Body Dynamics” (June, 9–13, 2007) [in Russian], Donetsk, Inst. of Appl. Math. and Mech., Ukrainian Nat. Acad. Sci. (2007), pp. 81–82.Google Scholar
  190. 190.
    M. V. Shamolin, “Complete integrability cases in dynamics on tangent bundle of two-dimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007).MathSciNetGoogle Scholar
  191. 191.
    M. V. Shamolin, “Complete integrability of equations of motion of a spatial pendulum in a medium flow taking account of rotational derivatives of its interaction force moment,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 3, 187–192 (2007).Google Scholar
  192. 192.
    M. V. Shamolin, Methods for Analyzing Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007).Google Scholar
  193. 193.
    M. V. Shamolin, “The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” Proc. 9th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2007), Lodz, Poland, December 17–20, 2007, Vol. 1, Tech. Univ. Lodz (2007), pp. 415–422.Google Scholar
  194. 194.
    M. V. Shamolin, “A three-parameter family of phase portraits in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 418, No. 1, 146–51 (2008).MathSciNetGoogle Scholar
  195. 195.
    S. Smale, “Rough systems are not dense,” in: Mathematics (collection of translations) [in Russian], 11, No. 4, 107–112 (1967).Google Scholar
  196. 196.
    S. Smale, “Differentiable dynamical systems,” Usp. Mat. Nauk, 25, No. 1, 113–185 (1970).MathSciNetGoogle Scholar
  197. 197.
    V. A. Steklov, On Rigid Body Motion in a Fluid [in Russian], Khar’kov (1893).Google Scholar
  198. 198.
    S. Sternberg, Lectures on Differential Geometry [Russian translation], Mir, Moscow (1970).Google Scholar
  199. 199.
    L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in Soliton Theory [in Russian], Nauka, Moscow (1986).Google Scholar
  200. 200.
    S. J. Takiff, “Rings of invariant polynomials for class of Lie algebras,” Trans. Amer. Math. Soc., 160, 249–262 (1971).MATHMathSciNetCrossRefGoogle Scholar
  201. 201.
    Ph. Tondeur, “Affine Zuzammenhänge auf Mannigfaltingkeiten mit fast-symplectischer Structur,” Comment. Math. Helv., 36, No. 1, 234–244 (1961).MATHMathSciNetCrossRefGoogle Scholar
  202. 202.
    H. H. Torriani, Extensions of Simple Lie Algebras, Integrable Systems of Toda Lattice Type and the Heat Equation, Preprint Univ. de Saõ Paulo RT-MAP-9001 (1990).Google Scholar
  203. 203.
    V. V. Trofimov, “Euler equations on Borel subalgebras of semisimple Lie algebras,” Izv. Akad. Nauk SSSR, Ser. Mat., 43, No. 3, 714–732 (1979).MATHMathSciNetGoogle Scholar
  204. 204.
    V. V. Trofimov, “Finite-dimensional representations of Lie algebras and completely integrable systems,” Mat. Sb., 111, No. 4, 610–621 (1980).MathSciNetGoogle Scholar
  205. 205.
    V. V. Trofimov, “Group-theoretic interpretation of equations of magnetic hydrodynamics of ideally conducting fluid,” Nelin. Koleb. Teor. Upravl., No. 3, 118–124 (1981).Google Scholar
  206. 206.
    V. V. Trofimov, “Completely integrable geodesic flows of left-invariant metrics on Lie groups related to commutative graded algebras with Poincaré duality,” Dokl. Akad. Nauk SSSR, 263, No. 4, 812–816 (1982).MathSciNetGoogle Scholar
  207. 207.
    V. V. Trofimov, “Completely integrable system of hydrodynamic type and algebras with Poincaré duality,” in: School in Operator Theory on Function Spaces [in Russian], Minsk (1982), pp. 192–193.Google Scholar
  208. 208.
    V. V. Trofimov, “Commutative graded algebras with Poincaré duality and Hamiltonian systems,” in: Topological and Geometric Methods in Mathematical Physics [in Russian], Voronezhsk. Univ., Voronezh (1983), pp. 128–132.Google Scholar
  209. 209.
    V. V. Trofimov, “Extensions of Lie algebras and Hamiltonian systems,” Izv. Akad. Nauk SSSR, Ser. Mat., 47, No. 6, 1303–1328 (1983).MATHMathSciNetGoogle Scholar
  210. 210.
    V. V. Trofimov, “Group-theoretic interpretation of some classes of equations of classical mechanics,” in: Differential Equations and Their Applications [in Russian], Izd. Mosk. Univ., Moscow (1984), pp. 106–111.Google Scholar
  211. 211.
    V. V. Trofimov, “A new method for constructing completely integrable Hamiltonian systems,” in: Qualitative Theory of Differential Equations and Control Theory of Motion [in Russian], Saransk (1985), pp. 35–39.Google Scholar
  212. 212.
    V. V. Trofimov, “Flat symmetric spaces with noncompact motion groups and Hamiltonian systems,” Tr. Sem. Vekt. Tenz. Anal., No. 22, 163–174 (1985).Google Scholar
  213. 213.
    V. V. Trofimov, “Geometric invariants of completely integrable Hamiltonian systems,” in: Proc. of All-Union Conf. in Geometry “in the large” [in Russian], Novosibirsk (1987), p. 121.Google Scholar
  214. 214.
    V. V. Trofimov, “On the geometric properties of the complete commutative set of functions on symplectic manifold,” in: Abstract of Baku International Topological Conference, Baku (1987), p. 297.Google Scholar
  215. 215.
    V. V. Trofimov, “Generalized Maslov classes of Lagrangian surfaces in symplectic manifolds,” Usp. Mat. Nauk, 43, No. 4, 169–170 (1988).Google Scholar
  216. 216.
    V. V. Trofimov, “Maslov index of Lagrangian submanifolds of symplectic manifolds,” Tr. Sem. Vekt. Tenz. Anal., No. 23, 190–194 (1988).Google Scholar
  217. 217.
    V. V. Trofimov, “On Fomenko conjecture for totally geodesic submanifolds in symplectic manifolds with almost Kählerian metric,” in: Selected Problems of Algebra, Geometry and Discrete Mathematics [in Russian], Moscow (1988), pp. 122–123.Google Scholar
  218. 218.
    V. V. Trofimov, “Geometric invariants of Lagrangian foliations,” Usp. Mat. Nauk, 44, No. 4, 213 (1989).Google Scholar
  219. 219.
    V. V. Trofimov, Introduction to Geometry of Manifolds with Symmetries [in Russian], Izd. Mosk. Univ., Moscow (1989).Google Scholar
  220. 220.
    V. V. Trofimov, “On geometric properties of a complete involutive family of functions on a symplectic manifold,” in: Baku Int. Topological Conf. [in Russian], Baku (1989), pp. 173–184.Google Scholar
  221. 221.
    V. V. Trofimov, “On the connection on symplectic manifolds and the topological invariants of Hamiltonian systems on Lie algebras,” in: Abstracts of Int. Conf. in Algebra, Novosibirsk (1989), p. 102.Google Scholar
  222. 222.
    V. V. Trofimov, “Symplectic connections, Maslov index, and Fomenko conjecture,” Dokl. Akad. Nauk SSSR, 304, No. 6, 1302–1305 (1989).Google Scholar
  223. 223.
    V. V. Trofimov, “Connections on manifolds and new characteristic classes,” Acta Appl. Math., 22, 283–312 (1991).MATHMathSciNetCrossRefGoogle Scholar
  224. 224.
    V. V. Trofimov, “Generalized Maslov classes and cobordisms,” Tr. Sem. Vekt. Tenz. Anal., No. 24, 186–198 (1991).Google Scholar
  225. 225.
    V. V. Trofimov, “Holonomy group and generalized Maslov classes of submanifolds in affine connection spaces,” Mat. Zametki, 49, No. 2, 113–123 (1991).MathSciNetGoogle Scholar
  226. 226.
    V. V. Trofimov, “Maslov index in pseudo-Riemannian geometry,” in: Algebra, Geometry and Discrete Mathematics in Nonlinear Problems [in Russian], Moscow (1991), pp. 198–203.Google Scholar
  227. 227.
    V. V. Trofimov, “Symplectic connections and Maslov–Arnold characteristic classes,” Adv. Sov. Math., 6, 257–265 (1991).MathSciNetGoogle Scholar
  228. 228.
    V. V. Trofimov, “Flat pseudo-Riemannian structure on tangent bundle of a flat manifold,” Usp. Mat. Nauk, 47, No. 3, 177–178 (1992).MathSciNetGoogle Scholar
  229. 229.
    V. V. Trofimov, “Path space and generalized Maslov classes of Lagrangian submanifolds,” Usp. Mat. Nauk, 47, No. 4, 213–214 (1992).MATHMathSciNetGoogle Scholar
  230. 230.
    V. V. Trofimov, “Pseudo-Euclidean structure of zero index on tangent bundle of a flat manifold,” in: Selected Problems of Algebra, Geometry, and Discrete Mathematics [in Russian], Moscow (1992), pp. 158–162.Google Scholar
  231. 231.
    V. V. Trofimov, “On absolute parallelism connections on a symplectic manifold,” Usp. Mat. Nauk, 48, No. 1, 191–192 (1993).MATHMathSciNetGoogle Scholar
  232. 232.
    V. V. Trofimov and A. T. Fomenko, “Dynamical systems on orbits of linear representations and complete integrability of some hydrodynamic systems,” Funkts. Anal. Prilozh., 17, No. 1, 31–39 (1983).MathSciNetGoogle Scholar
  233. 233.
    V. V. Trofimov and A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras,” Usp. Mat. Nauk, 39, No. 2, 3–56 (1984).MathSciNetGoogle Scholar
  234. 234.
    V. V. Trofimov and A. T. Fomenko, “Geometry of Poisson brackets and methods for Liouville integrability of systems on symmetric spaces,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Novejsh. Distizh., 29, 3–108 (1986).MathSciNetGoogle Scholar
  235. 235.
    V. V. Trofimov and A. T. Fomenko, “Geometric and algebraic mechanisms of integrability of Hamiltonian systems on homogeneous spaces,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 16, 227–299 (1987).MathSciNetGoogle Scholar
  236. 236.
    V. V. Trofimov and M. V. Shamolin, “Dissipative systems with nontrivial generalized Arnol’d–Maslov classes,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 62 (2000).Google Scholar
  237. 237.
    M. B. Vernikov, “To definition of connections concordant with symplectic structure,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 7, 77–79 (1980).Google Scholar
  238. 238.
    J. Vey, “Deformation du crochet de Poisson sur une varieté symplectique,” Comment. Math. Helv., 50, No. 4, 421–454 (1975).MATHMathSciNetCrossRefGoogle Scholar
  239. 239.
    S. V. Vishik and S. F. Dolzhanskii, “Analogs of Euler–Poisson equations and magnetic hydrodynamics equations related to Lie groups,” Dokl. Akad. Nauk SSSR, 238, No. 5, 1032–1035.Google Scholar
  240. 240.
    M. Y. Wang, “Parallel spinors and parallel forms,” Ann. Global Anal. Geom., 7, No. 1, 59–68 (1989).MATHMathSciNetCrossRefGoogle Scholar
  241. 241.
    A. Weinstein, “Local structure of Poisson manifolds,” J. Differential Geom., 18, No. 3, 523–558 (1983).MATHMathSciNetGoogle Scholar
  242. 242.
    J. Wolf, Spaces of Constant Curvature [Russian translation], Nauka, Moscow (1982).Google Scholar

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© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia

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