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Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems. (English. Russian original) Zbl 1353.37121

J. Math. Sci., New York 180, No. 4, 365-530 (2012); translation from Fundam. Prikl. Mat. 16, No. 4, 3-229 (2010).
Summary: 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 doctoral 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).

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37N05 Dynamical systems in classical and celestial mechanics
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
Full Text: DOI

References:

[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).
[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).
[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).
[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).
[5] A. A. Andronov, A Collection of Works [in Russian], USSR Academy of Sciences, Moscow (1956).
[6] A. A. Andronov and E. A. Leontovich, ”Some cases of dependence of limit cycles on a parameter,” Uch. Zap. GGU, No. 6 (1937).
[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). · JFM 64.1402.04
[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).
[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). · Zbl 0133.35102
[10] A. A. Andronov and E. A. Leontovich, ”Dynamical systems of the first nonroughness degree on the plane,” Mat. Sb., 68, No. 3 (1965). · Zbl 0143.11902
[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).
[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). · Zbl 0168.06801
[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)
[14] A. A. Andronov and L. S. Pontryagin, ”Rough systems,” Dokl. Akad. Nauk SSSR, 14, No. 5, 247–250 (1937).
[15] D. V. Anosov, ”Geodesic flows on closed Riemannian manifolds of negative curvature,” Tr. Mat. Inst. Akad. Nauk SSSR, 90 (1967). · Zbl 0176.19101
[16] S. Kh. Aronson and V. Z. Grines, ”Topological classification of flows on closed two-dimensional manifolds,” Usp. Mat. Nauk, 41, No. 1 (1986)
[17] V. I. Arnol’d, ”On characteristic class entering quantization conditions,” Funkts. Anal. Prilozh., 1, No. 1, 1–14 (1967). · Zbl 0175.20303 · doi:10.1007/BF01075861
[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).
[19] V. I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989).
[20] V. I. Arnol’d and A. B. Givental’, ”Symplectic geometry,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 4, 5–139 (1985).
[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).
[22] D. Arrowsmith and C. Place, Ordinary Differential Equations, Qualitative Theory with Applications [Russian translation], Mir, Moscow (1986). · Zbl 0671.34001
[23] G. F. Baggis, ”Rough systems of two differential equations,” Usp. Mat. Nauk, 10, No. 4 (1955).
[24] E. A. Barashin and V. A. Tabueva, Dynamical Systems with Cylindrical Phase Space [in Russian], Nauka, Moscow (1969).
[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). · Zbl 0059.08201
[26] N. N. Bautin, ”Some methods for qualitative studying dynamical systems related to field turn,” Prikl. Mat. Mekh., 37, No. 6 (1973). · Zbl 0296.34019
[27] N. N. Bautin and E. A. Leontovich, Methods and Tools for Qualitative Studying Dynamical Systems on Plane [in Russian], Nauka, Moscow (1976). · Zbl 0785.34004
[28] I. Bendixon, ”On curves defined by differential equations,” Usp. Mat. Nauk, 9 (1941).
[29] M. Berger, Geometry [Russian translation], Mir, Moscow (1984).
[30] A. L. Besse, Manifolds All of Whose Geodesics Are Closed [Russian translation], Mir, Moscow (1981).
[31] J. Birkhoff, Dynamical Systems [Russian translation], Gostekhizdat, Moscow (1941).
[32] J. A. Bliess, Lectures on the Calculus of Variations [Russian translation], Gostekhizdat, Moscow–Leningrad (1941).
[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).
[34] O. I. Bogoyavlenskii, ”Some integrable cases of Euler equations,” Dokl. Akad. Nauk SSSR, 287, No. 5, 1105–1108 (1986).
[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).
[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).
[37] N. Bourbaki, Integration [Russian translation], Nauka, Moscow (1970).
[38] N. Bourbaki, Lie Groups and Algebras [Russian translation], Mir, Moscow (1972). · Zbl 0249.22001
[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).
[40] A. V. Brailov, ”Some cases of complete integrability of Euler equations and applications,” Dokl. Akad. Nauk SSSR, 268, No. 5, 1043–1046 (1983). · Zbl 0544.58011
[41] A. D. Bryuno, Local Method for Nonlinear Analysis of Differential Equations [in Russian], Nauka, Moscow (1979). · Zbl 0496.34002
[42] V. S. Buslaev and V. A. Nalimova, ”Trace formula in Lagrangian Mechanics,” Teor. Mat. Fiz., 61, No. 1, 52–63 (1984). · Zbl 0598.70023 · doi:10.1007/BF01038547
[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).
[44] N. N. Butenina, ”To bifurcation theory of dynamical systems under field turn,” Differ. Uravn., 10, No. 7 (1974).
[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). · Zbl 0626.43003 · doi:10.1007/BF01077990
[46] S. A. Chaplygin, Selected Works, [in Russian], Nauka, Moscow (1976).
[47] Dao Chiog Tkhi and A. T. Fomenko, Minimal Surfaces and Plateau Problem [in Russian], Nauka, Moscow (1987). · Zbl 0617.53001
[48] P. Dazord, ”Invariants homotopiques attachés aux fibrés simplectiques,” Ann. Inst. Fourier (Grenoble), 29, No. 2, 25–78 (1979). · Zbl 0378.58011 · doi:10.5802/aif.743
[49] R. Delanghe, F. Sommen, and V. Sousek, Clifford Algebra and Spinor-Valued Functions, Kluwer Academic, Dordrecht (1992).
[50] F. M. Dimentberg, Theory of Spatial Hinge Mechanisms [in Russian], Nauka, Moscow (1982). · Zbl 0513.93051
[51] J. Dixmier, Universal Enveloping Algebras [Russian translation], Mir, Moscow (1978).
[52] C. T. G. Dodson, Categories, Bundles and Spacetime Topology, Kluwer Academic, Dordrecht (1988). · Zbl 0661.53016
[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). · Zbl 0780.58019
[54] B. A. Dubrovin and S. P. Novikov, ”On Poisson brackets of hydrodynamic type,” Dokl. Akad. Nauk SSSR, 279, No. 2, 294–297 (1984). · Zbl 0591.58012
[55] B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry, Methods and Applications [in Russian], Nauka, Moscow (1986).
[56] A. T. Fomenko, Symplectic Geometry [in Russian], Izd. Mosk. Unuv., Moscow (1988).
[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). · Zbl 0746.58037
[58] A. T. Fomenko and D. B. Fuks, A Course of Homotopical Topology [in Russian], Nauka, Moscow (1989). · Zbl 0675.55001
[59] A. T. Fomenko and V. V. Trofimov, Integrable Systems on Lie Algebras and Symmetric Spaces, Gordon and Breach, New York (1988). · Zbl 0659.58018
[60] H. Fujimoto, ”Examples of complete minimal surfaces in $ {\(\backslash\)mathbb{R}\^m} $ whose Gauss maps omit $ \(\backslash\)frac{{m\(\backslash\)left( {m + 1} \(\backslash\)right)}}{2} $ hyperplanes in general positions,” Sci. Rep. Kanazawa Univ., 33, No. 2, 37–43 (1989). · Zbl 0673.53033
[61] D. B. Fuks, ”On Maslov–Arnol’d characteristic classes,” Dokl. Akad. Nauk SSSR, 178, No. 2, 303–306 (1968). · Zbl 0175.20304
[62] D. V. Georgievskii and M. V. Shamolin, ”Kinematics and geometry of masses of a rigid body with a fixed point in $ {\(\backslash\)mathbb{R}\^n} $ ,” Dokl. Ross. Akad. Nauk, 380, No. 1, 47–50 (2001).
[63] D. V. Georgievskii and M. V. Shamolin, ”On kinematics of a rigid body with a fixed point in $ {\(\backslash\)mathbb{R}\^n} $ . Abstracts of sessions of workshop ’Actual Problems of Geometry and Mechanics’,” Fundam. Prikl. Mat., 7, No. 1, 315 (2001).
[64] D. V. Georgievskii and M. V. Shamolin, ”Generalized dynamical Euler equations for a rigid body with a fixed point in $ {\(\backslash\)mathbb{R}\^n} $ ,” Dokl. Ross. Akad. Nauk, 383, No. 5, 635–637 (2002).
[65] D. V. Georgievskii and M. V. Shamolin, ”First integrals of equations of motion of generalized gyroscope in $ {\(\backslash\)mathbb{R}\^n} $ ,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 37–41 (2003). · Zbl 1127.70003
[66] D. V. Georgievskii and M. V. Shamolin, ”Valerii Vladimirovich Trofimov,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 23, 5–6 (2007). · Zbl 1153.01325
[67] D. V. Georgievskii and M. V. Shamolin, ”On kinematics of a rigid body with a fixed point in $ {\(\backslash\)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).
[68] D. V. Georgievskii and M. V. Shamolin, ”Generalized dynamical Euler equations for a rigid body with a fixed point in $ {\(\backslash\)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).
[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).
[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).
[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).
[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).
[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).
[74] V. V. Golubev, Lectures on Analytic Theory of Differential Equations [in Russian], Gostekhizdat, Moscow–Leningrad (1950).
[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).
[76] G. V. Gorr, L. V. Kudryashova, and L. A. Stepanova, Classical Problems of Rigid Body Dynamics [in Russian], Naukova Dumka, Kiev (1978).
[77] D. N. Goryachev, ”New cases of integrability of dynamical Euler equations,” Varshav. Univ. Izvestiya, Book 3, 1–15 (1916). · JFM 48.1428.01
[78] M. De Gosson, ”La définition de l’indice de Maslov sans hypothèse de transversalité,” C. R. Acad. Sci. Paris, 310, 279–282 (1990). · Zbl 0705.22012
[79] M. Goto and F. Grosshans, Semisimple Lie Algebras [Russian translation], Mir, Moscow (1981). · Zbl 0528.17001
[80] P. Griffiths, Exterior Differential forms and Calculus of Variations [Russian translation], Mir, Moscow (1986).
[81] D. M. Grobman, ”On homeomorphism of systems of differential equations,” Dokl. Akad. Nauk SSSR, 128, No. 5, 880–881 (1959). · Zbl 0100.29804
[82] D. M. Grobmam, ”Topological classification of neighborhoods of a singular point in n-dimensional space,” Mat. Sb., 56, No. 1, 77–94 (1962).
[83] M. Gromov, ”Pseudo-holomorphic curves in symplectic manifolds,” Invent. Math., 82, 307–347 (1985). · Zbl 0592.53025 · doi:10.1007/BF01388806
[84] V. Guillemin and S. Sternberg, Geometric Asymptotics [Russian translation], Mir, Moscow (1981).
[85] V. Guillemin and S. Sternberg, Symplectic Technique in Physics, Cambridge Univ. Press, Cambridge (1984). · Zbl 0576.58012
[86] Ph. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).
[87] R. Harvey and H. B. Lawson, ”Calibrated geometry,” Acta Math., 148, 47–157 (1982). · Zbl 0584.53021 · doi:10.1007/BF02392726
[88] S. Helgason, Differential Geometry and Symmetric Spaces [Russian translation], Mir, Moscow (1964). · Zbl 0122.39901
[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.
[90] M. D. Hvidsten, ”Volume and energy stability for isometric minimal immersions,” Illinois J. Math., 33, No. 3, 488–494 (1989). · Zbl 0661.53040
[91] M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets, Geometry and Quantization [in Russian], Nauka, Moscow (1991). · Zbl 0731.58002
[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.
[93] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 1, Nauka, Moscow (1981). · Zbl 0526.53001
[94] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 2, Nauka, Moscow (1981). · Zbl 0526.53001
[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.
[96] A. I. Kostrikin, An Introduction to Algebra [in Russian], Nauka, Moscow (1977). · Zbl 0464.00007
[97] V. V. Kozlov, Methods of Qualitative Analysis in Rigid Body Dynamics [in Russian], Izd. Mosk. Univ., Moscow (1980). · Zbl 0557.70009
[98] V. V. Kozlov, ”Integrability and nonintegrability in Hamiltonian mechanics,” Usp. Mat. Nauk, 38, No. 1, 3–67 (1983).
[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). · Zbl 0422.70022
[100] L. D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media [in Russian], Nauka, Moscow (1982).
[101] S. Lang, Introduction to Theory of Differentiable Manifolds [Russian translation], Mir, Moscow (1967). · Zbl 0146.17503
[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).
[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. · Zbl 0745.53034
[104] Le Hong Van and A. T. Fomenko, ”Minimality criterion of Lagrangian submanifolds in Kälerian manifolds,” Mat. Zametki, 46, No. 4, 559–571 (1987). · Zbl 0633.53082
[105] S. Leftschets, Geometric Theory of Differential Equations [Russian translation], IL, Moscow (1961).
[106] V. G. Lemlein, ”On spaces of symmetric and almost symmetric connection,” Dokl. Akad. Nauk SSSR, 116, No. 4, 655–658 (1957). · Zbl 0080.37801
[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).
[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).
[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).
[110] Yu. I. Levi, ”On affine connections adjacent to a skew-symmetric tensor,” Dokl. Akad. Nauk SSSR, 128, No. 4, 668–671 (1959).
[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. · Zbl 0486.58015
[112] J. Lion and M. Vergne, Weyl Representation, Maslov Index, and Theta-Series [Russian translation], Mir, Moscow (1983). · Zbl 0542.22012
[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.
[114] Yu. I. Manin, ”Algebraic aspects of nonlinear differential equations,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 11, 5–112 (1978).
[115] O. V. Manturov, ”Homogeneous spaces and invariant tensors,” Itogi Nauki Tekh. Ser. Probl. Geom., 18, 105–142 (1986).
[116] J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York (1976). · Zbl 0346.58007
[117] J. Marsden and A. Weinstein, ”Reduction of symplectic manifolds with symmetry,” Rep. Math. Phys., 5, No. 1, 121–130 (1974). · Zbl 0327.58005 · doi:10.1016/0034-4877(74)90021-4
[118] V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Izd. Mosk. Univ., Moscow (1965). · Zbl 0653.35002
[119] V. P. Maslov, Asymptotic Methods and Perturbation Theory [in Russian], Nauka, Moscow (1988). · Zbl 0653.35002
[120] W. Miller, Symmetry and Separation of Variables [Russian translation], Mir, Moscow (1981).
[121] R. Miron and V. Oproiu, ”Almost cosymplectic and conformal almost cosymplectic connections,” Revue Roum. Math. Pures Appl., 16, No. 6, 893–912 (1971). · Zbl 0217.18903
[122] A. S. Mishchenko, B. Yu. Sternin, and V. E. Shatalov, Lagrangian Manifolds and Canonical Operator Method [in Russian], Nauka, Moscow (1978). · Zbl 0437.58007
[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). · Zbl 0383.58006
[124] J. M. Morvan, ”Classes de Maslov d’une immersion Lagrangienne et minimalit’e,” C. R. Acad. Sci. Paris, 292, 633–636 (1981). · Zbl 0466.53030
[125] Yu. I. Neimark, ”On motions close to double-asymptotic motion,” Dokl. Akad. Nauk SSSR, 172, No. 5, 1021–1024 (1957).
[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.
[127] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow–Leningrad (1949).
[128] Z. Nitetski, Introduction to Differential Dynamics [Russian translation], Mir, Moscow (1975).
[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. · Zbl 0833.58019
[130] S. P. Novikov, ”Hamiltonian formalism and a multivalued analog of Morse theory,” Usp. Mat. Nauk, 37, No. 5, 3–49 (1982).
[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).
[132] J. Palais and S. Smale, ”Structural stability theorems,” in: Mathematics (collection of translations) [in Russian], 13, No. 2, 145–155 (1969).
[133] J. Palis and W. De Melu, Geometric Theory of Dynamical Systems. An Introduction [Russian translation], Mir, Moscow (1986).
[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). · Zbl 0357.17004 · doi:10.1063/1.522992
[135] M. Peixoto, ”On structural stability,” Ann. Math. (2), 69, 199–222 (1959). · Zbl 0084.08403 · doi:10.2307/1970100
[136] M. Peixoto, ”Structural stability on two-dimensional manifolds,” Topology, 1, No. 2, 101–120 (1962). · Zbl 0107.07103 · doi:10.1016/0040-9383(65)90018-2
[137] M. Peixoto, ”On an approximation theorem of Kupka and Smale,” J. Differ. Equ., 3, 214–227 (1966). · Zbl 0153.40901 · doi:10.1016/0022-0396(67)90026-5
[138] V. A. Pliss, ”On roughness of differential equations given on torus,” Vestn. Leningr. Univ. Mat., 13, 15–23 (1960). · Zbl 0096.29404
[139] V. A. Pliss, Integral Sets of Periodic Systems of Differential Equations [in Russian], Nauka, Moscow (1967). · Zbl 0189.39601
[140] H. Poincaré, On Curves Defined by Differential Equations [Russian translation], OGIZ, Moscow–Leningrad (1947).
[141] H. Poincaré, ”New methods in celestial mechanics,” in: Selected Works [Russian translation], Vols. 1. 2, Nauka, Moscow (1971, 1972). · Zbl 0232.01018
[142] H. Poincaré, On Science [Russian translation], Nauka, Moscow (1983).
[143] A. G. Reiman, ”Integrable Hamiltonian systems related to graded Lie algebras,” Zap. Nauchn. Sem. LOMI Akad. Nauk SSSR, 95, 3–54 (1980).
[144] B. L. Reinhart, ”Cobordism and Euler number,” Topology, 2, 173–178 (1963). · Zbl 0178.26402 · doi:10.1016/0040-9383(63)90031-4
[145] S. T. Sadetov, ”Integrability conditions for Kirchhoff equations,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 56–62 (1990).
[146] S. Salamon, Riemannian Geometry and Holonomy Group Research Notes Math. Ser., Vol. 201, Wiley, New York (1989). · Zbl 0685.53001
[147] E. J. Saletan, ”Contraction of Lie groups,” J. Math. Phys., 2, No. 1, 1–22 (1961). · Zbl 0098.25804 · doi:10.1063/1.1724208
[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).
[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). · Zbl 0705.70008
[150] R. Schoen and S. T. Yau, ”Complete three dimensional manifolds with positive Ricci curvature and scalar curvature,” Ann. Math. Stud., 209–228 (1982). · Zbl 0481.53036
[151] H. Seifert and W. Threlfall, Topology [Russian translation], Gostekhizdat, Moscow (1938).
[152] J. P. Serre, ”Singular homologies of fiber spaces,” in: Fiber Spaces and Applications [Russian translation], IL, Moscow (1958), pp. 9–114.
[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). · Zbl 0753.70007
[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). · Zbl 0753.70007
[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).
[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).
[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). · Zbl 0815.34002
[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). · Zbl 0855.70014
[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).
[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). · Zbl 0874.70006 · doi:10.4213/rm940
[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). · Zbl 0923.70008
[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). · Zbl 0900.70152
[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).
[164] M. V. Shamolin, ”Spatial Poincaré topographical systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997). · Zbl 0915.58062 · doi:10.4213/rm859
[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).
[166] M. V. Shamolin, ”On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998). · Zbl 0925.34003 · doi:10.4213/rm53
[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).
[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). · Zbl 1065.70500
[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). · Zbl 0968.34039 · doi:10.4213/rm217
[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).
[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). · Zbl 0968.34039
[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).
[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).
[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).
[175] M. V. Shamolin, ”On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000). · Zbl 0968.34021 · doi:10.4213/rm305
[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).
[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).
[178] M. V. Shamolin, ”Integrability cases of equations of spatial rigid body dynamics,” Prikl. Mekh., 37, No. 6, 74–82 (2001). · Zbl 1010.70520
[179] M. V. Shamolin, ”On integration of some classes of nonconservative systems, Usp. Mat. Nauk, 57, No. 1, 169–170 (2002). · Zbl 1054.34503 · doi:10.4213/rm489
[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). · Zbl 1006.34035 · doi:10.1023/A:1015026512786
[181] M. V. Shamolin, ”Foundations of differential and topological diagnostics,” J. Math. Sci., 114, No. 1, 976–1024 (2003). · Zbl 1067.93020 · doi:10.1023/A:1021807110899
[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). · Zbl 1067.70006 · doi:10.1023/A:1021865626829
[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). · Zbl 1140.70456 · doi:10.1023/B:JOTH.0000029572.16802.e6
[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). · Zbl 1116.74378
[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).
[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). · Zbl 1100.74546
[187] M. V. Shamolin, ”On a certain integrability case of equations of dynamics on $ {\(\backslash\)text{so}}(4) \(\backslash\)times {\(\backslash\)mathbb{R}\^n} $ ,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005). · doi:10.4213/rm1688
[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.
[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.
[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). · Zbl 1137.37325 · doi:10.4213/rm7487
[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).
[192] M. V. Shamolin, Methods for Analyzing Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007). · Zbl 1334.70001
[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.
[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). · Zbl 1257.70015
[195] S. Smale, ”Rough systems are not dense,” in: Mathematics (collection of translations) [in Russian], 11, No. 4, 107–112 (1967).
[196] S. Smale, ”Differentiable dynamical systems,” Usp. Mat. Nauk, 25, No. 1, 113–185 (1970). · Zbl 0205.54201
[197] V. A. Steklov, On Rigid Body Motion in a Fluid [in Russian], Khar’kov (1893).
[198] S. Sternberg, Lectures on Differential Geometry [Russian translation], Mir, Moscow (1970). · Zbl 0211.53501
[199] L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in Soliton Theory [in Russian], Nauka, Moscow (1986). · Zbl 0632.58003
[200] S. J. Takiff, ”Rings of invariant polynomials for class of Lie algebras,” Trans. Amer. Math. Soc., 160, 249–262 (1971). · Zbl 0232.22027 · doi:10.1090/S0002-9947-1971-0281839-9
[201] Ph. Tondeur, ”Affine Zuzammenhänge auf Mannigfaltingkeiten mit fast-symplectischer Structur,” Comment. Math. Helv., 36, No. 1, 234–244 (1961). · Zbl 0103.38901 · doi:10.1007/BF02566901
[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).
[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).
[204] V. V. Trofimov, ”Finite-dimensional representations of Lie algebras and completely integrable systems,” Mat. Sb., 111, No. 4, 610–621 (1980).
[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).
[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).
[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.
[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.
[209] V. V. Trofimov, ”Extensions of Lie algebras and Hamiltonian systems,” Izv. Akad. Nauk SSSR, Ser. Mat., 47, No. 6, 1303–1328 (1983). · Zbl 0547.58024
[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.
[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.
[212] V. V. Trofimov, ”Flat symmetric spaces with noncompact motion groups and Hamiltonian systems,” Tr. Sem. Vekt. Tenz. Anal., No. 22, 163–174 (1985). · Zbl 0604.58030
[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.
[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.
[215] V. V. Trofimov, ”Generalized Maslov classes of Lagrangian surfaces in symplectic manifolds,” Usp. Mat. Nauk, 43, No. 4, 169–170 (1988).
[216] V. V. Trofimov, ”Maslov index of Lagrangian submanifolds of symplectic manifolds,” Tr. Sem. Vekt. Tenz. Anal., No. 23, 190–194 (1988). · Zbl 0808.53036
[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. · Zbl 0738.57012
[218] V. V. Trofimov, ”Geometric invariants of Lagrangian foliations,” Usp. Mat. Nauk, 44, No. 4, 213 (1989). · Zbl 0659.17009
[219] V. V. Trofimov, Introduction to Geometry of Manifolds with Symmetries [in Russian], Izd. Mosk. Univ., Moscow (1989). · Zbl 0705.53002
[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. · Zbl 0850.46025
[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.
[222] V. V. Trofimov, ”Symplectic connections, Maslov index, and Fomenko conjecture,” Dokl. Akad. Nauk SSSR, 304, No. 6, 1302–1305 (1989). · Zbl 0687.58009
[223] V. V. Trofimov, ”Connections on manifolds and new characteristic classes,” Acta Appl. Math., 22, 283–312 (1991). · Zbl 0732.58018 · doi:10.1007/BF00580851
[224] V. V. Trofimov, ”Generalized Maslov classes and cobordisms,” Tr. Sem. Vekt. Tenz. Anal., No. 24, 186–198 (1991). · Zbl 0967.37501
[225] V. V. Trofimov, ”Holonomy group and generalized Maslov classes of submanifolds in affine connection spaces,” Mat. Zametki, 49, No. 2, 113–123 (1991). · Zbl 0737.53036
[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. · Zbl 0818.57015
[227] V. V. Trofimov, ”Symplectic connections and Maslov–Arnold characteristic classes,” Adv. Sov. Math., 6, 257–265 (1991). · Zbl 0780.57017
[228] V. V. Trofimov, ”Flat pseudo-Riemannian structure on tangent bundle of a flat manifold,” Usp. Mat. Nauk, 47, No. 3, 177–178 (1992). · Zbl 0778.53019
[229] V. V. Trofimov, ”Path space and generalized Maslov classes of Lagrangian submanifolds,” Usp. Mat. Nauk, 47, No. 4, 213–214 (1992). · Zbl 0792.58016
[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. · Zbl 0818.53084
[231] V. V. Trofimov, ”On absolute parallelism connections on a symplectic manifold,” Usp. Mat. Nauk, 48, No. 1, 191–192 (1993). · Zbl 0808.53035
[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). · Zbl 0521.58031
[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). · Zbl 0549.58024
[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).
[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). · Zbl 0797.58006
[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).
[237] M. B. Vernikov, ”To definition of connections concordant with symplectic structure,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 7, 77–79 (1980). · Zbl 0445.53013
[238] J. Vey, ”Deformation du crochet de Poisson sur une varieté symplectique,” Comment. Math. Helv., 50, No. 4, 421–454 (1975). · Zbl 0351.53029 · doi:10.1007/BF02565761
[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.
[240] M. Y. Wang, ”Parallel spinors and parallel forms,” Ann. Global Anal. Geom., 7, No. 1, 59–68 (1989). · Zbl 0688.53007 · doi:10.1007/BF00137402
[241] A. Weinstein, ”Local structure of Poisson manifolds,” J. Differential Geom., 18, No. 3, 523–558 (1983). · Zbl 0524.58011
[242] J. Wolf, Spaces of Constant Curvature [Russian translation], Nauka, Moscow (1982).
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