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MR2373767 37J35 (34C40) Shamolin, M. V. (RS-MOSC) A case of complete integrability in dynamics on a tangent bundle of a two-dimensional sphere. (Russian) Uspekhi Mat. Nauk 62 (2007), no. 5(377), 169-170; translation in Russian Math. Surveys 62 (2007), no. 5, 1009-1011. {A review for this item is in process.} c Copyright American Mathematical Society 2008

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MR2342525 58A05 (57Rxx) A idagulov, R. R. (RS-MOSCM) ; Shamolin, M. V. (RS-MOSCM) Manifolds of continuous structures. (Russian. Russian summary) Sovrem. Mat. Fundam. Napravl. 23 (2007), 71-86. {A review for this item is in process.} c Copyright American Mathematical Society 2008

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MR2342524 74Bxx (35Q72) A idagulov, R. R. (RS-MOSCM) ; Shamolin, M. V. (RS-MOSCM) A general spectral approach to the dynamics of a continuous medium. (Russian. Russian summary) Sovrem. Mat. Fundam. Napravl. 23 (2007), 52-70. {A review for this item is in process.} c Copyright American Mathematical Society 2008


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MR2342523 (Review) 22A30 A idagulov, R. R. (RS-MOSCM) ; Shamolin, M. V. (RS-MOSCM) Archimedean uniform structures. (Russian. Russian summary) Sovrem. Mat. Fundam. Napravl. 23 (2007), 46-51. Summary (translated from the Russian): "In mathematics, the concept of the Archimedean property is used in connection with two different objects: orderings of groups and valuations of rings. In both cases, one can define a topology on these objects and even a uniform structure; in the first case, an interval topology, and in the second, a certain valuation. It turns out that these two uses of the term Archimedean property and the somewhat regrettable term `topological group without small subgroups' are special cases of the concept of the Archimedean property of a topological group." c Copyright American Mathematical Society 2008

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MR2342521 01A70 Georgievski D. V. ; Shamolin, M. V. i, Valeri Vladimirovich Trofimov. (Russian) i Sovrem. Mat. Fundam. Napravl. 23 (2007), 5-15. {There will be no review of this item.} c Copyright American Mathematical Society 2008

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MR2252203 (2007c:70009) 70E15 (70E40 70E45 70H06) Shamolin, M. V. Comparison of Jacobi-integrable cases of two- and three-dimensional motions of a body in a medium in the case of a jet flow. (Russian. Russian summary) Prikl. Mat. Mekh. 69 (2005), no. 6, 1003-1010; translation in J. Appl. Math. Mech. 69 (2005), no. 6, 900-906 (2006). Summary (translated from the Russian): "We show the complete integrability of the plane problem of the motion of a rigid body in a resisting medium under jet flow conditions, when one first integral, which is a transcendental function of quasi-velocities (in the sense of the theory of functions of a complex variable with essentially singular points), exists in the system of equations of motion. It is assumed that the entire interaction of the medium with the body is concentrated on a part of the surface of the body that has the shape of a (one-dimensional) plate. We generalize this plane problem to the three-dimensional case, where a complete set of first integrals exists for the equations of motion: one analytic, one meromorphic, and one transcendental. Here we assume that the entire interaction of the medium with the body is concentrated on part of the surface of the body that has the shape of a flat (two-dimensional) disk. We also attempt to construct a generalization of the `low-dimensional' cases to the dynamics of a so-called four-dimensional rigid body whose interaction with a medium is concentrated on a part of the (three-dimensional) surface of the body that has the shape of a (three-dimensional) sphere. In this case, the angular velocity vector is six-dimensional, while the velocity of the center of mass is four-dimensional." c Copyright American Mathematical Society 2007, 2008

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MR2225204 (2007a:70009) 70E40 (37J35 70H06) Shamolin, M. V. (RS-MOSC) On an integrable case of equations of dynamics on so(4) з R4 . (Russian) Uspekhi Mat. Nauk 60 (2005), no. 6(366), 233-234; translation in Russian Math. Surveys 60 (2005), no. 6, 1245-1246. The four-dimensional analog of the problem on the motion of a rigid body under the action of resistance forces with variable dissipation and of a servo-constraint is studied. The rotational part of the equations of motion is considered under the assumption that the body is dynamically symmetric. It is shown that under appropriate conditions the equations of motion possess an invariant surface. For the motions restricted to this surface transcendental first integrals are indicated. Reviewed by Alexander Burov References


1. O. I. Bogoyavlenskii, Dokl. Akad. Nauk SSSR 287 (1986), 1105-1108; English transl., Soviet Phys. Dokl. 31 (1986), 309-311. MR0839710 (87j:70005) 2. O. I. Bogoyavlenskii and G. F. Ivakh, Uspekhi Mat. Nauk 40:4 (1985), 145-146; English transl., Russian Math. Surveys 40:4 (1985), 161-162. MR0807729 (87g:58035) 3. M. V. Shamolin, J. Math. Sci. (New York) 114 (2003), 919-975. MR1965083 (2004d:70008) 4. M. V. Shamolin, Dokl. Akad. Nauk 375 (2000), 343-346; English transl., Dokl. Phys. 45:11 (2000), 632-634. MR1833828 (2002c:70005) 5. V. V. Trofimov and A. T. Fomenko, Itogi Nauki i Tekhniki: Sovremennye Problemy Mat.: Fundamental'nye Napravleniya, vol. 29, VINITI, Moscow 1987, pp. 3-108; English transl., J. Soviet Math. 39 (1987), 2683-2746. MR0892743 (88i:58059) 6. S. A. Chaplygin, Selected works, Nauka, Moscow 1976. (Russian) MR0424502 (54 #12464) 7. M. V. Shamolin, Dokl. Akad. Nauk 364 (1999), 627-629; English transl., Dokl. Phys. 44:2 (1999), 110-113. MR1702618 (2000k:70008)
Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

c Copyright American Mathematical Society 2007, 2008

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MR2216035 (2006m:70012) 70E40 Shamolin, M. V. (RS-MOSC-IMC) A case of complete integrability in the three-dimensional dynamics of a rigid body interacting with a medium taking into account the rotational derivatives of the momentum of forces of angular velocity. (Russian) Dokl. Akad. Nauk 403 (2005), no. 4, 482-485; translation in Dokl. Phys. 50 (2005), no. 8, 414-418. From the text (translated from the Russian): "Because of its complexity, the problem of the motion of a rigid body in an unbounded medium requires the introduction of simplifying restrictions. The main goal is to introduce hypotheses that allow one to study the motion of a rigid body separately from the motion of the medium in which the body is located. On the one hand, a similar approach was taken in the classical Kirchhoff problem of the motion of a body in an unbounded ideal incompressible fluid which is at rest at infinity and which undergoes irrotational motion. On the other hand, it is clear that the aforementioned Kirchhoff problem does not exhaust the possibilities of this type of modeling. "In this paper, we consider the possibility of transferring the results of the dynamics of the planeparallel motion of a homogeneous axisymmetric rigid body interacting at its front circular face with a uniform flow of a resisting medium to the case of three-dimensional motion. Here, unlike in previous papers on the modeling of the interaction between a medium and a rigid body, we take


into account the effects of the so-called rotational derivatives of the moment of hydroaerodynamic forces with respect to the components of the angular velocity of the rigid body itself." c Copyright American Mathematical Society 2006, 2008

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MR2171058 (2006d:11150) 11Y16 A idagulov, R. R. ; Shamolin, M. V. A refinement of Conway's algorithm. (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2005, no. 3, 53-55, 71; translation in Moscow Univ. Math. Bull. 60 (2005), no. 3, 34-35 (2006). Summary (translated from the Russian): "We refine Conway's algorithm for computing prime numbers. In the course of analyzing it, we establish that some numbers obtained using it are erroneous. Further investigation leads to the determination of tabularly computable functions and the establishment of the equivalence of this class of functions and the class of recursive functions." c Copyright American Mathematical Society 2006, 2008

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MR2131714 (2005m:70050) 70E99 (70K99) Shamolin, M. V. (RS-MOSC-MC) Geometric representation of motion in a problem of the interaction of a body with a medium. (Russian. English, Ukrainian summaries) Prikl. Mekh. 40 (2004), no. 4, 137-144; translation in Internat. Appl. Mech. 40 (2004), no. 4, 480-486. From the text (translated from the Russian): "We carry out a complete qualitative analysis of a model version of the plane-parallel motion of a body in a resisting medium under a jet-flow condition for the oscillatory domain of the phase space of the dynamic equations. We assume that the velocity of the center of the plate through which the body interacts with the medium remains constant throughout the motion. As was shown earlier, the dynamical system in the space of quasivelocities is relatively structurally stable (robust with respect to physically admissible classes of dynamical systems). We consider an additional qualitative integration of the kinematic relations. We study the properties of the solutions corresponding to the oscillatory domain: the properties of the asymptotes associated with the motion of the rigid body, various equivalence relations in the


trajectory space, topological analogies, and mechanical interpretations of asymptotic motions. We study the local property of the asymptote." c Copyright American Mathematical Society 2005, 2008

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MR2082898 (2005j:70014) 70E99 (34C40 37N05 70K05) Shamolin, M. V. Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body. Dynamical systems. J. Math. Sci. (N. Y.) 122 (2004), no. 1, 2841-2915. In this paper the author applies a rather simplified mechanical model for the study and explanation of nontrivial effects arising in the plane-parallel and spatial motions of a rigid body in a resisting medium. He assumes that the interaction of the medium with the body is concentrated at the front part of the body's surface, which has the form of a flat plate. The paper consists of six chapters. In the first chapter the author considers several forms of nonlinear dynamical systems describing the motion of a body in a medium. In the next four chapters the author develops the standard methods of the qualitative theory of ordinary differential equations and successfully uses them for investigation and classification of phase trajectories of dissipative systems of special types. In the final chapter he studies the stability of the translational deceleration and self-oscillations of the moving body in the presence of a linear damping momentum. Reviewed by A. Ya. Savchenko References 1. Hubert Airy, "The soaring of birds," Nature, XXVIII, 1.596. 2. G. A. Al'ev, "Three-dimensional problem on disk immersion into a compressible fluid," Izv. Akad. Nauk SSSR Mekh. Zhid. Gaza, No. 1, 17-20 (1988). 3. V. V. Amel'kin, N. A. Lukashevich, and A. P. Sadovskii, Nonlinear Oscillations in SecondOrder Systems [in Russian], Belorus. Gos. Univ., Minsk (1982). MR0670589 (84a:34037) 4. A. A. Andronov, Collection of Works [in Russian], Moscow, Izd. Akad. Nauk SSSR (1956). 5. A. A. Andronov and E. A. Leontovich, "Certain cases of dependency of limit cycles on the parameter," Uchenye Zapiski Gork. Gos. Univ., Issue 6 (1937). 6. A. A. Andronov and E. A. Leontovich, "On the theory of changes in the qualitative structure of the partition of a plane into trajectories," Dokl. Akad. Nauk SSSR, 21, Issue 9 (1938). 7. A. A. Andronov and E. A. Leontovich, "Bifurcation of limit cycles from a structurally unstable focus or center and from a structurally unstable limit cycle," Mat. Sb., 40, No. 2 (1956). MR0085413 (19,36a)


8. A. A. Andronov and E. A. Leontovich, "On the bifurcation of limit cycles from a separatrix loop and from the separatrix of the equilibrium state of the saddle-node type," Mat. Sb., 48, No. 3, 179-224 (1959). MR0131612 (24 #A1461) 9. A. A. Andronov and E. A. Leontovich, "Dynamical systems of first degree of structural iInstability on the plane," Mat. Sb., 68, No. 3, 328-372 (1965). MR0194657 (33 #2866) 10. A. A. Andronov, and E. A. Leontovich, "Sufficient conditions for the first-degree structural instability of a dynamical system on the plane," Differ. Uravn., 6, No. 12, 2121-2134 (1970). MR0279399 (43 #5121) 11. A. A. Andronov and L. S. Pontryagin, "Rough systems," Dokl. Akad. Nauk SSSR, 14, No. 5, 247-250 (1937). 12. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillations [in Russian], Nauka, Moscow (1981). MR0665745 (83i:34002) 13. A. A. Andronov, E. A. Leontovich, I. I, Gordon, and A. G. Mayer, The Qualitative Theory of Second-Order Dynamical Systems [in Russian], Nauka, Moscow (1966). 14. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Mayer, Theory of Bifurcations of Dynamical Systems on the Plane [in Russian], Nauka, Moscow (1967). MR0235228 (38 #3539) 15. D. V. Anosov, "Geodesic flows on closed Riemannian manifolds of negative curvature," Tr. Mat. Inst. Steklova 90, Nauka, Moscow (1967). MR0224110 (36 #7157) 16. P. Appel, Theoretical Mechanics (in Two Volumes) [in Russian], Fizmatgiz, Moscow (1960). 17. S. Kh. Aranson, "Dynamical systems on two-dimensional manifolds,"In: Proc. 5th Internat. Conf. on Nonlinear Oscillations, Vol 2, [in Russian], Inst. Mat. Akad. Nauk Ukr. SSR, Kiev (1970), pp. 46-52. 18. S. Kh. Aranson and V. Z. Grines, "Topological classification of flows on closed two-dimensional manifolds," Usp. Mat. Nauk, 41, No. 1, 149-169 (1986). MR0832412 (87j:58075) 19. Yu. A. Arkhangel'skii, Analytical Dynamics of Rigid Bodies [in Russian], Nauka, Moscow (1977). 20. V. I. Arnold, "The Euler equations of dynamics of rigid bodies in the ideal fluid are Hamiltonian," Usp. Mat. Nauk, 24, No. 3, 225-226 (1969). MR0277163 (43 #2900) 21. V. I. Arnold, Supplementary Chapters of Theory of Ordinary Dufferential Equations [in Russian], Nauka, Moscow (1978). MR0526218 (80i:34001) 22. V. I. Arnold, Ordinary Differential Equations [in Russian], Nauka, Moscow (1984). MR0799024 (86i:34001) 23. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989). MR1037020 (93c:70001) 24. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics [in Russian], VINITI, Moscow (1985). 25. D. Arrowsmitt and C. Place, Ordinary Differential Equations. A Qualitative Approch with Applications [Russian transl.], Mir, Moscow (1986) (Originally published: London: Chapman and Hall, 1982). MR0684068 (85a:34001) 26. I. M. Babakov, Theory of Oscillations [in Russian], Nauka, Moscow (1965). MR0187444 (32 #4894)


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69. N. N. Butenina, "To the theory of bifurcations of dynamical systems under field rotation," Differ. Uravn., 10, No. 7, 1322-1324 (1974). MR0357960 (50 #10425) 70. N. N. Butenina, "On the possibility of rotation of the vector field of a dynamical system through an angle with passage through systems of first degree of roughness only," In: Theory of Oscillations, Applied Mathematics and Cybernetics. Interschool Collection of Works, Gor'ky (1974), pp. 15-28. 71. G. S. Byushgens and R. V. Studnev, The Dynamics of Longitudinal and Lateral Motions [in Russian], Mashinostroenie, Moscow (1969). 72. G. S. Byushgens and R. V. Studnev, The Dynamics of an Aircraft. Three-Dimensional Motion [in Russian], Mashinostroenie (1988). 73. M. L. Byalyi, "On polynomial in momentum first integrals for the mechanical system on a two-dimensional torus," Funkts. Anal. Ego Prilozh., 21, No. 4, 64-65 (1987). MR0925074 (89c:70029) 74. S. A. Chaplygin, "On the motion of massive bodies in an incompressible fluid," In: Complete Collection of Works [in Russian], Vol. 1, Izd. Akad. Nauk SSSR, Leningrad (1933), pp. 133- 135. 75. S. A. Chaplygin. Selected Works [in Russian], Nauka, Moscow (1976). MR0424503 (54 #12465) 76. F. L. Chernous'ko, L. D. Akulenko, and B. N. Sokolov, Control of Vibrations [in Russian], Nauka, Moscow (1980). 77. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations [Russian transl.], Inostr. Lit., Moscow (1958). 78. S. A. Dovbysh, "Intersection of asymptotic surfaces of the perturbed Euler-Poinsot equations," Prikl. Mat. Mekh., 51, No. 3, 363-370 (1987). MR0960031 (89g:70005) 79. S. A. Dovbysh, "Separatrix splitting and Birth of Isolated Periodical Solutions in One-and-aHalf Degree of Freedom Hamiltonian Systems," Usp. Mat. Nauk, 44, No. 44, 229-230 (1989). MR0998366 (90i:58140) 80. S. A. Dovbysh, "Splitting of separatrices of unstable uniform rotatiopns and the nonitegrability of the Lagrange perturbed problem," Vestn. Moskov. Gos. Univ. Ser I. Mat. Mekh., No. 3, 70-77 (1990). MR1064299 (92d:70005) 81. B. A. Dubrovin and S. N. Novikov, "On Poisson brackets of the hydrodynamic type," Dokl. Akad. Nauk SSSR, 279, No. 2, 294-297 (1984). MR0770656 (86e:58021) 82. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979). MR0566582 (81f:53001) 83. G. Dyulak, On Limit Cycles [in Russian], Nauka, Moscow (1980). MR0597517 (82k:34031) 84. G. E. Dzhakal'ya, Perturbation Theory Methods for Linear Systems [in Russian], Nauka, Moscow (1967). 85. V. A. Eroshin, "Penetration of a cone into a liqiud layer," Vestn. Moskov. Gos. Univ. Ser I. Mat. Mekh., No. 5, 53-59 (1963). MR0155493 (27 #5427) 86. V. A. Eroshin, "Ricochets of Plates from the Surface of Ideal Incompressible Fluid," Vestn. Moskov. Gos. Univ. Ser. I. Mat. Mekh., No. 6, 99-104 (1970). 87. V. A. Eroshin, "The immersion of a disk in a compressible fluid at an angle to the free surface,"


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a resisting medium," In: VII Chetaev Conf. on Analytical Mechanics, Motion Stability and Control (June 10-13, 1997, Kazan'). Abstracts of Reports [in Russian], Izd. Kazansk. Gos Univ., Kazan' (1997), p. 24. V. A. Samsonov, V. A. Eroshin, G. A. Konstantinov, and V. M. Makarshin, Two Model Problems of Motion of a Body in a Medium in the Jet Flow Past This Body. Research Report of the Institute of Mechanics of Moscow State University No. 3427 [in Russian], Moscow (1987). V. A. Samsonov, B. Ya. Lokshin, and V. A. Privalov, Qualitative Analysis. Research Report of the Institute of Mechanics of Moscow State University No. 3245 [in Russian], Moscow (1985). V. A. Samsonov, M. V. Shamolin, V. A. Eroshin, and V. M. Makarshin, Mathematical Modeling in the Problem of the Deceleration of a Body in a Resisting Medium in the Jet Flow Past this Body. Research Report of the Institute of Mechanics of Moscow State University No. 4396 [in Russian], Moscow (1995). G. Sansonet, Ordinary Differential Equations [Russian transl.], Inostr. Liter., Moscow (1954). L. I. Sedov, Continuum Mechanics [in Russian] Vol. 1, Nauka, Moscow (1983); Vol. 2, Nauka (1984). MR0434068 (55 #7037) G. Seifert and V. Trelfall, Topology [Russian transl.], Gostekhizdat, Moscow-Leningrad (1938). M. V. Shamolin, Qualitative Analysis of the Model Problem on Motion of a Body in a Medium in Jet Flow. Thesis [in Russian], Mosk. Gos. Univ., Moscow (1991). M. V. Shamolin, "Closed trajectories of distinct topological types in the problem of motion of a body in a resisting medium," Vestn. Moskov. Gos. Univ. Ser. I Mat., Mekh., No. 2, 52-56, 112 (1992). MR1293705 (95d:34060) M. V. Shamolin, "On the problem of motion of a body in a resisting medium," Vestn. Moskov. Gos. Univ. Ser. I Mat., Mekh., No. 1, 52-58,112 (1992). MR1214592 (93k:70028) M. V. Shamolin, "Classification of phase portraits in the problem of motion of a body in a resisting medium in the presence of linear damping moment," Prikl. Mat. Mekh., 57, Issue 4, 40-49 (1993). MR1258007 (94i:70027) M. V. Shamolin, "A new two-parameter family of phase portraits in a problem on the motion of a body in a resisting medium," In: Modeling and Investigaton of System Stability. Scientific Conf. Kiev, May 24-28, 1993. Abstracts of Reports. Part 2 [in Russian], Znanie, Kiev (1993), pp. 62-63. M. V. Shamolin, "Relative Structural Stability in the Problem on the Motion of a Body in a Resisting Medium," In: Mechanics and Its Applications. Scientific Conf. Tashkent, November 9-11 1993. Abstracts of Reports [in Russian], Tashkentsk Gos. Univ., Tashkent (1993), pp. 20-21. Д M. V. Shamolin, "Application of Poincare map systems and reference systems in some particular systems of differential equations," Vestn. Moskov. Gos. Univ. Ser. I Mat., Mekh., No. 2, 66-70, 113 (1993). MR1223987 (94b:34060) M. V. Shamolin, "Existence and uniqueness of trajectories having infinitely remote points as limit sets for dynamical systems on the plane," Vestn. Moskov. Gos. Univ. Ser. I Mat., Mekh., No. 1, 68-71, 112 (1993). MR1293942 (95e:34036) M. V. Shamolin, "New two-parameter family of phase portraits in the problem of motion of a body in a medium," Dokl. Ross. Akad. Nauk, 337, No. 5, 611-614 (1994). MR1298329


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(95g:70006) M. V. Shamolin, "On relative roughness of dynamical systems in the problem on the motion of a body in a medium," In: Modeling and Investigation of System Stability. Scientific Conf. Kiev, May 16-20. Abstracts of Reports [in Russian], Kiev, (1994), pp. 144-145. M. V. Shamolin, "A new two-parameter family of phase portraits with limit cycles in the dynamics of a rigid body interacting with a medium," In: Modeling and Study of System Stability. Scientific Conf. Kiev, May 15-19, 1995. Abstracts of Reports (System Study) [in Russian], Kiev (1995), p. 125. M. V. Shamolin, "Relative structural stability of dynamical systems in the problem on the motion of a body in a medium," In: B. E. Pobedri and V. V. Kozlov (ed.),. Analytical, Numerical and Experimental Methods in Mechanics. Collection of Research Papers [in Russian], Mosk. Gos. Univ., Moscow (1995), pp. 14-19. MR1809236 M. V. Shamolin, "On the relative roughness of dynamical systems in the problem on the motion of a body in a resisting medium. Abstract of report at Chebyshev readings." Vestn. Moskov. Gos. Univ. Ser. I Mat. Mekh., No. 6, 17, (1995). MR1809236 M. V. Shamolin, "The definition of relative structural stability and a two-parameter family of phase portraits in the dynamics of a rigid body," Usp. Mat. Nauk, 51, Issue 1, 175-176 (1996). MR1392692 (97f:70010) M. V. Shamolin, "Periodic and Poisson stable trajectories in the problem of motion of a body in a resisting medium," Izv. Ross. Akad. Nauk Mekh. Tv. Tela, No. 2, 55-63 (1996). Д M. V. Shamolin, "Spatial topographical Poincare systems and reference systems," In: The Erugin Readings III. Matem. Conf. Brest, May 14-16, 1996. Abstracts of Reports, Brest (1996), p. 107. M. V. Shamolin, "Introduction to three-dimensional dynamics of motion of a rigid body in a resisting medium," In: Proc. Intern. Conf. and Chebyshev Readings on the Occasion of P. L. Chebyshev's 175th Anniversary (Moscow, May 14-19, 1996). Vol. 2 [in Russian], Mosk. Gos. Univ., Moscow (1996), pp. 371-373. M. V. Shamolin, "A list of integrals for dynamic equations in the three-dimensional problem on the motion of a body in a resisting medium," In: Modeling and Investigation of System Stability. Scientific Conf. (Kiev, May 20-24, 1996). Abstracts of Reports (System Study) [in Russian], Kiev (1996), p. 142. M. V. Shamolin, "A variety of types of phase portraits in the dynamics of a rigid body interacting with a resisting medium," Dokl. Ross. Akad. Nauk, 349, No. 2, 193-197 (1996). MR1440994 (98b:70009) M. V. Shamolin, "Qualitative methods in the dynamics of a rigid body interacting with medium," In: II Sibir' Congress on Applied and Industrial Mechanics (Novosibirsk, June 25-30, 1996). Abstracts of Papers. Part III [in Russian], Novosibirsk (1996), p. 267. M. V. Shamolin, "On an integrable case in the dynamics of the three-dimensional motion of a body in a resisting medium," In: II Symposium on Classical and Celestial Mechanics (Velikie Luki, August 23-28, 1996). Abstracts of Reports [in Russian], Moscow-Velikie Luki (1996), p. 91-92. M. V. Shamolin, "Introduction to the problem on the deceleration of a body in a resisting


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255. 256.

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262. 263.

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medium and a new two-parametric family of phase portraits," Vestn. Moskov. Gos. Univ. Ser. I Mat., Mekh., No. 4, 57-69 (1996). MR1644665 (99e:70027) M. V. Shamolin, "On the integrable case in the three-dimensional dynamics of a rigid body interacting with a medium, Izv. Ross. Akad. Nauk Mekh. Tv. Tela, No. 2, 65-68 (1997). M. V. Shamolin, "Jacobi integrability of the problem of a three-dimensional pendulum placed into the incoming flow of a medium," In:Modeling and Investigation of System Stability. Scientific Conf. (Kiev, May 19-23, 1997). Abstracts of Papers (Mechanical Systems) [in Russian], Kiev (1997), p. 143. M. V. Shamolin, "Partial Stabilization of Rotational Motions of a Body in a Medium in Free Deceleration," In: All-Russia Intern. Conf. on Problems of Celestial Mechanics (St. Petersburg, June 3-6, 1997), Izd. Inst. Teor. Astronomii Ross. Akad. Nauk (1997). Д M. V. Shamolin, "Spatial Poincare map systems and reference systems," Usp. Mat. Nauk, 52, Issue 3, 177-178 (1997). MR1479402 (99a:34089) M. V. Shamolin, "Mathematical modeling of dynamics of a three-dimensional pendulum with a medium flowing around it," In: Proceedings of VII Intern. Symposium on Methods of Discrete Singularities in Problems of Mathematical Physics (Feodosiya-Kherson, June 26-29, 1997) [in Russian], Izd. Kherson Gos. Tekhn. Univ., Kherson (1997), pp. 153-154. M. V. Shamolin, "Three-dimensional dynamics of a rigid body interacting with a medium," In: Workshop on Mechanical Systems, Motion Control and Navigation Problems, Izv. Ross. Akad. Nauk Mekh. Tverd. Tela, No. 4, 174 (1997). M. V. Shamolin, "Qualitative methods in the dynamics of a rigid body interacting with a medium," In: YSTM'96. Proceedings of Intern. Congress. Vol. 2. [in Russian], Moscow (1997), pp. 1-4. M. V. Shamolin, "Qualitative and numerical methods in certain problems of three-dimensional dynamics of a rigid body interacting with a medium," In: The 5th International Conf. on Engineering and Physical Problems of New Technology (Moscow, May 19-22, 1998) [in Russian], Mosk. Gos. Tekhn. Univ., Moscow (1998), pp. 154-155. M. V. Shamolin, "Certain problems of three-dimensional dynamics of a rigid body interacting with a medium under quasi-stationarity conditions," In: All-Russia Scientific Conf. of Young Scientists on Modern Problems of Aerospace Science (Zhukovskii, May 27-29, 1998) [in Russian], Izd. Tsentr. Aerogidrodin. Inst., Moscow (1998), pp. 89-90. M. V. Shamolin, "Absolute and relative structural stability in the three-dimensional dynamics of a rigid body interacting with a medium," Intern. Conf. on Industrial Mathematics (ICIM'98, Taganrog, June 29-July 3, 1998) [in Russian], Izd. Taganrogsk. Gos. Politekhn. Inst., Taganrog (1998), pp. 332-333. M. V. Shamolin, "On integrability in transcendental functions," Usp. Mat. Nauk, 53, Issue 3, 209-210 (1998). MR1657632 (99h:34006) M. V. Shamolin, "Families of three-dimensional phase portraits in the three-dimensional dynamics of a rigid body interacting with a medium," In: III Intern. Symp. on Classical and Celestial Mechanics (Velikie Luki, August 23-27, 1998). Abstracts of Papers [in Russian], Vych. Tsentr Ross. Akad. Nauk, Moscow-Velikie Luki (1998), pp. 165-167. M. V. Shamolin, "Methods of nonlinear analysis in dynamics of a rigid body interacting with


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a medium," In: CD-Proc. of the Cong. on Nonlinear Analysis and its Applications (Moscow, Sept. 1-5, 1998), Moscow (1998), pp. 497-508. M. V. Shamolin, "A family of portraits with limit cycles in the plane dynamics of a rigid body interacting with a medium," Izv. Ross. Akad. Nauk Mekh. Tv. Tela, No. 6, 29-37 (1998). M. V. Shamolin, "Certain classes of partial solutions in the dynamics of a rigid body interacting with a medium," Izv. Ross. Akad. Nauk Mekh. Tv. Tela, No. 2, 178-189 (1999). M. V. Shamolin, "New Jacobi integrable cases in the dynamics of a rigid body interacting with a medium," Dokl. Ross. Akad. Nauk, 364, No. 5, 627-629 (1999). MR1702618 (2000k:70008) M. V. Shamolin, "On the roughness of dissipative systems and relative roughness and nonroughness of systems with variable dissipation," Usp. Mat. Nauk, 54, No. 5, 181-182 (1999). MR1741681 (2000j:37021) M. V. Shamolin, "A new family of phase portraits in the three-dimensional dynamics of a rigid body interacting with a medium," Dokl. Ross. Akad. Nauk, 371, No. 4, 480-483 (2000). MR1776307 (2001k:70006) M. V. Shamolin, "On roughness of dissipative systems and relative roughness of variable dissipation systems. Abstract of paper at the Rashevskii seminar on vector and tensor analysis," Vestn. Moskov. Gos. Univ. Ser. I Mat. Mekh., No. 2, 63 (2000). M. V. Shamolin, "On limit sets of differential equations near singular points," Usp. Mat. Nauk, 375, No. 3, 187-188 (2000). MR1777365 (2002d:34049) M. V. Shamolin, "The Jacobi integrability in the problem of motion of a four-dimensional rigid body in a resisting medium," Dokl. Ross. Akad. Nauk, 375, No. 3, 343-346 (2000). MR1833828 (2002c:70005) M. V. Shamolin, "On the stability of the motion of a rigid body twisted about its longitudinal axis," Izv. Ross. Akad. Nauk Mekh. Tv. Tela, No. 1, 189-193 (2000). M. V. Shamolin, "Complete integrability of equations of motion of a three-dimensional pendulum in an incoming flow of a medium," Vestn. Moskov. Gos. Univ. Ser. I Mat. Mekh., No. 5, 22-28 (2000). MR1868040 (2002f:70005) M. V. Shamolin, "Global qualitative analysis of nonlinear systems on the problem of motion of a body in a resisting medium," In: Fourth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, August 18-21, 1993, Szeged, Hungary (1993), p. 54. M. V. Shamolin, "Relative structural stability on the problem of motion of a body in a resisting medium," In: ICM'94, Abstract of Short Communications, Zurich, 3-11 August, 1994, Zurich, Switzerland (1994), p. 207. M. V. Shamolin, "Structural optimization of controlled rigid motion in a resisting medium," In: WCSMO-1, Extended Abstracts. Posters, Goslar, May 28-June 2, 1995, Goslar, Germany (1995) pp. 18-19. MR1809236 M. V. Shamolin, "Qualitative methods for the dynamic model of interaction of a rigid body with a resisting medium and new two-parametric families of phase portraits," In: DynDays'95 (Sixteenth Annual Informal Workshop), Program and Abstracts, Lyon, June 28-July 1, 1995, Lyon, France (1995), p. 185. M. V. Shamolin, "New two-parameter families of phase patterns on the problem of motion of a body in a resisting medium," In: ICIAM'95, Book of Abstracts, Hamburg, 3-7 July, 1995,


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Hamburg, Germany (1995), p. 436. M. V. Shamolin, "Poisson-stable and dense orbits in rigid body dynamics," In: The 3rd Experimental Chaos Conference, Advance Program, Edinburg, Scotland, August 21-23, 1995, Edinburg, Scotland (1995), p. 114. M. V. Shamolin, "Qualitative methods in the dynamics of a rigid body interacting with a medium," In: Abstracts of GAMM Wissenschaftliche Jahrestangung'96, 27-31 May, 1996, Karls-Universitat Prag, Prague (1996), pp. 129-130. M. V. Shamolin, "Relative structural stability and relative structural instability of different degrees in topological dynamics," In: Abstracts of International Topological Conference Dedicated to P. S. Alexandroff 's 100th Birthday "Topology and Applications", Moscow, May 27-31, 1996, Phasys, Moscow (1996), pp. 207-208. M. V. Shamolin, "Topographical Poincare systems in higher-dimensional spaces," In: Fifth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, Regional Committee of the Hungarian Academy of Sciences, July 29-August 2, 1996, Szeged, Hungary (1996), p. 45. M. V. Shamolin, "Qualitative methods in the dynamics of a rigid body interacting with a medium," In: Abstracts of XIX ICTAM, Kyoto, Japan, August 25-31, 1996, Kyoto, Japan, (1996), p. 285. M. V. Shamolin, "Three-dimensional structural optimization of controlled rigid motion in a resisting medium," In: Proceedings of WCSMO-2, Zakopane, Poland, May 26-30, 1997, Zakopane, Poland (1997), p. 387-392. M. V. Shamolin, "Classical problem of three-dimensional motion of a pendulum in a jet flow," In: The 3rd EUROMECH Solid Mechanics Conference, Book of Abstracts, Stockholm, Sweden, August 18-22, 1997, Royal Inst. of Technology, Stockholm, Sweden (1997), p. 204. M. V. Shamolin, "Families of three-dimensional phase portraits in the dynamics of a rigid body," In: EQUADIFF 9, Abstracts, Enlarged Abstracts, Brno, Czech Rep., August 25-29, 1997, Masaryk Univ., Brno, Czech Rep. (1997), p. 76. Д M. V. Shamolin, "Higher-dimensional topographical Poincare systems in rigid body dynamics," In: Abstracts of GAMM Wissenschaftliche Jahrestangung'98, 6-9 April, 1998, Universitat Bremen, Bremen, Germany (1998), p. 128. M. V. Shamolin, "New two-parametric families of phase portraits in three-dimensional rigid body dynamics," In: Intern. Conf. Dedicated to 90th Birthday of L. S. Pontryagin, Moscow, August 31-September 9, 1998. Abstracts of Reports. Differential Equations, Izd. Moskov. Gos. Univ., Moscow (1998), pp. 97-99. Д M. V. Shamolin, "Lyapunov functions method and higher-dimensional Poincare topographical systems in rigid body dynamics," In: IV Crimea Intern. Mat. School. The Lyapunov Function Method and Its Applications. Abstracts of Reports. Crimea. Alushta (05-12.09. 1998), Izd. Simpheropolsk. Gos. Univ., Sympheropol' (1998), p. 80. M. V. Shamolin, "Some classical problems in the three-dimensional dynamics of a rigid body interacting with a medium," In: Proc. of ICTACEM'98, Kharagpur, India, Dec.1-5, 1998, Aerospace Engineering Dep., Indian Inst. of Technology, Kharagpur, India (1998) 11 p. (CDROM, Printed at Printek Point, Technology Market, KGP-2).


292. M. V. Shamolin, "Integrability in Terms of Transcendental Functions in Rigid Body Dynamics," In: Book of Abstracts of GAMM Annual Meeting, April 12-16 1999, Metz, France, Universite de Metz, Metz (1999), p. 144. 293. M. V. Shamolin and D. V. Shabarshov, " Lagrange Tori and the Hamilton-Jacobi Equation," In: Book of Abstracts of Conference PDE Prague'98 (Praha, August 10-16, 1998; (Partial Differential Equations: Theory and Numerical Solutions), Charles University, Praha, Czech Rep (1998), p. 88. 294. M. V. Shamolin and S. V. Tsypstyn, Analytical and Numerical Study of Trajectories of a Body Moving in a Resisting Medium. Research Report of the Institute of Mechanics of Moscow State University No. 4289. [in Russian], Moscow (1993). 295. M. V. Shamolin and D. V. Shebarshov, "Projections of Lagrangian tori for a biharmonic oscillator onto the position space and dynamics of a rigid body interacting with a medium," in: Modeling and Investigation of System Stability. Scientific Conf. (Kiev, May 19-23, 1997). Abstracts of Papers (Mechanical Systems) [in Russian], Kiev (1997), p. 142. 296. O. P. Shorygin and N. A. Shul'man, "Disk entry into water with an attack angle," Uch. Zap. TsAGI, 8, No. 1, 12-21 (1977). 297. S. Smale, "Rough systems are not dense," Sb. Per. Mat., 11, No. 4, 107-112 (1967). 298. S. Smale, "Differentiable dynamical systems," Usp. Mat. Nauk, 25, No. 1, 113-185 (1970). MR0263116 (41 #7721) 299. V. M. Starzhinskii, Applied Methods of Nonlinear Vibrations [in Russian], Nauka, Moscow (1977). MR0495355 (58 #14067) 300. V. A. Steklov, On the Motion of a Rigid Body in a Fluid [in Russian], Khar'kov (1893). 301. V. V. Stepanov, A Course in Differential Equations [in Russian], Fizmatgiz, Moscow (1959). 302. V. V. Strekalov, "Ricochet in the entry of a disk with a plane surface close to the vertical one into water," Uch. Zap. TsAGI, 8, No. 5, 66-73 (1977). 303. G. K. Suslov, Theoretical Mechanics [in Russian], Gostekhizdat, Moscow (1946). 304. V. V. Sychev, A. I. Ruban, Vik. V. Sychev, and G. L. Korolev, Asymptotic Theory of Detached Flows [in Russian], Nauka, Moscow (1987). 305. J. L. Synge, Classical Dynamics [Russian transl.], Fizmatgiz, Moscow (1963). 306. V. G. Tabachnikov, "Stationary characeristics of wings at low speeds over the whole range of angles of attack," Trudy Centr. Aerohydrodyn. Inst., Issue 1621, Moscow (1974), pp. 18-24. 307. Ya. V. Tatarinov, Lectures on Classical Dynamics, Izd. Moskov. Gos. Univ., Moscow (1984). MR0778381 (85m:70001) 308. V. V. Trofimov, "Embeddings of finite grioups as regular elements into compact Lie groups," Dokl. Akad. Nauk SSSR, 226, No. 4, 785-786 (1976). MR0439984 (55 #12865) 309. V. V. Trophimov, "Euler's equations on finite-dimensional solvable Lie groups," Izv. Akad. Nauk SSSR Ser. Mat., 44, No. 5, 1191-1199 (1980). MR0595263 (82e:70006) 310. V. V. Trofimov, "Symplectic structures on automorphism groups of symmetric spaces," Vestn. Moskov. Gos. Univ. Ser. I Mat. Mekh., No. 6, 31-33 (1984). MR0775300 (86b:53038) 311. V. V. Trofimov, "Geometric Invariants of Completely Integrable Systems," In: All-Union Conf. on Geometry "in the Large." Abstracts of Reports [in Russian], Novosibirsk (1987), p. 121. 312. V. V. Trofimov and A. T. Fomenko, "A Technique for constructing Hamiltonian flows on


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Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

c Copyright American Mathematical Society 2005, 2008

Citations Article From References: 0 From Reviews: 0


MR2042218 (2004j:70012) 70E45 (37J35 70H06) Georgievski D. V. ; Shamolin, M. V. i, First integrals of the equations of motion of a generalized gyroscope in Rn . (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2003, no. 5, 37-41, 72-73; translation in Moscow Univ. Math. Bull. 58 (2003), no. 5, 25-29 (2004). Summary (translated from the Russian): "By analogy with a three-dimensional space, a generalized gyroscope in Rn is a rigid body with a fixed point in which all the moments of inertia with respect to n hyperplanes are divided into two groups, and in each group the moments are equal to each other. In this case, the well-known system of n(n - 1)/2 generalized Euler dynamic equations has a specified number of first integrals, which depends on the inertia structure of the gyroscope, and reduces to a linear nonhomogeneous nonautonomous system. We study in detail the case n = 4." c Copyright American Mathematical Society 2004, 2008

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MR1965084 (2004d:93033) 93B30 (34H05 93C15) Shamolin, M. V. Foundations of differential and topological diagnostics. Dynamical systems, 12. J. Math. Sci. (N. Y.) 114 (2003), no. 1, 976-1024. The paper presents the results of an analysis of the aircraft control system diagnostics problem. The motion of the control system is described by nonlinear ordinary differential equations. First the diagnostics problem is considered in the case of exact trajectorial measurements. Second, trajectorial measurements corrupted by normal white noise with zero mean value and bounded spectrum are analyzed. Third, trajectorial measurements are investigated in the case when their errors are normal random variables whose absolute values are bounded by a certain function of time. Solving the diagnostics problem allows one to repair the control system and isolate the trouble. To solve the diagnostics problem the following data are used: the mathematical model of the motion of the object, the bounded domain of its initial conditions and the list of models describing the motion of this object with a fault. A statistical-modelling method for selection of the checking surface selection is proposed and checking surface accessibility conditions are investigated. A description of diagnostic techniques involving selection of the checking surface is proposed. Particular attention is given to the case of linear systems. The author emphasizes the deep differences of his approach to the diagnostics problem in comparison with other works. Namely, the diagnostics problem is considered in terms of a classification of malfunctions. The mathematical modelling of malfunctions is presented in this context. General statements of the


differential diagnostics problem are discussed in detail. Various extensions of the diagnostics theorem are described. Reviewed by Valery I. Korobov (Khar kov) References 1. P. P. Parkhomenko and E. S. Sagomonyan, Foundations of Engineering Diagnostics [in Russian], Energiya, Moscow (1981). 2. V. V. Karibskii, P. P. Parkhomenko, E. S. Sagomonyan, and V. F. Khalchev, Foundations of Engineering Diagnostics. Book 1 [in Russian], Energiya, Moscow (1976). 3. L. A. Mironovskii, "Functional diagnostics of dynamical systems," Avtom. Telemekh., No. 3, 96-121 (1980). 4. A. V. Maiorov, G. K. Moskatov, and G. P. Shibanov, Safety in Operation of Automated Plants [in Russian], Mashinostroenie, Moscow (1988). 5. B. N. Petrov, V. J. Rutkovskij, I. N. Krutova, and Zemlyakov S. D., Principles of Construction and Design of Self-Adapting Control Systems [in Russian], Mashinostroenie, Moscow (1972). 6. V. J. Rutkovskii, S D. Zemlyakov, V. M. Glumov, et al. "Adaptive algorithmic methods for diagnostics and faultless operation of control systems," In: Proc. 3rd IMECO Symposium on Technical Diagnostics, Moscow, 1983. Budapest, Publ. IMECO (1985), pp. 173-180. 7. I. T. Borisenok, "Redundant control systems," in Proc. 3rd All-Union Conf. on Control Systems (Engeneering Cybernetics) [in Russian], Odessa (1963). 8. I. T. Borisenok, "On the differential restorability theory. Certain control and stability problems for mechanical systems," Papers of the Institute of Mechanics of Lomonosov Moscow State University. Issue 22 [in Russian], Izd. Mosk. Gos. Univ., 101-108 (1973). 9. M. Eykhoff, Foundations of Control System Identification [Russian translation], Mir, Moscow (1975). 10. V. I. Belyakov and I. T. Borisenok. "Modeling methods using the list of probable flaws in functional diagnostics of dynamical systems," VIII Symp. on Problems of Data System Redundancy. Abstract of Papers. Leningrad (1983), Part 3. 11. V. I. Belyakov, I. T. Borisenok, and V. A. Samsonov, "On an algorithm for continuous expressdiagnostics," Avtomat. Telemekh., No. 3, 113-116 (1973). 12. V. I. Belyakov and I. T. Borisenok, "On a method for the construction of a checking surface in the diagnostic problem," Certain Problems of Dynamics of Controlled Rigid Bodies. Collection of Papers [in Russian], Izd. Mosk. Gos. Univ., Moscow (1982). 13. I. T. Borisenok and B. Ya. Lokshin, "On the problem of system diagnostics," All-Union Symp. on Diagnostic System Design. Abstracts of Papers [in Russian], Rostov-on-Don, May 1978. 14. I. T. Borisenok and I. A. Smarzhevskii, "Diagnostics of a certain directly controlled system," In: Readings in Memory of V. N. Chelomei. Abstracts of Papers [in Russian], Moscow, Institute of Engineering Science, USSR Academy of Sciences, June 1989. 15. Yu. M. Okunev and N. A. Parusnikov, Structural and Algorithmic Aspects of the Set-Up of Control Problems [in Russian], Izd. Mosk. Gos. Univ., Moscow (1983). 16. I. T. Borisenok and I. A. Smarzhevskii, Navigation and Control Theory for Moving Objects. Techniques for Functional Diagnostics of an Aircaft's Motion Control System. Research Report


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MR1965083 (2004d:70008) 70E40 (34C60 37N05 70F40 70G40) Shamolin, M. V. New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium. Dynamical systems, 12. J. Math. Sci. (N. Y.) 114 (2003), no. 1, 919-975. Several models are considered for the motion of a rigid body in a resisting medium (like a gas or a fluid). Probably the best known model in this area is the Kirchhoff system for the motion of a rigid body in an ideal fluid. In the present paper somewhat different physical assumptions are used. The resulting differential equations are nonlinear systems of dimension 2, 3, or 4. Their phase portraits are analysed, and several integrable cases are pointed out. Reviewed by Yuri B. Suris References 1. A. A. Andronov, Collection of Works [in Russian], Moscow, Izd. Akad. Nauk SSSR (1956). 2. A. A. Andronov and E. A. Leontovich, "Certain cases of the dependency of limit cycles on the parameter," Uchenye Zapiski Gork. Gos. Univ., No. 6 (1937). 3. A. A. Andronov and E. A. Leontovich, "On the theory of changes in the qualitative structure


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100. V. V. Nemytskii and V. V. Stepanov, The Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow-Leningrad (1949). 101. Z. Nitecki, Introduction to Differential Dynamics [Russian translation], Mir, Moscow (1975). MR0649789 (58 #31211) 102. S. P. Novikov and I. Shmel'tser, "Periodic solutions to the Kirchoff equations of free motion of a rigid body and the ideal fluid and the exended Lusternik-Shnirel'man-Morse (LSM) theory. I," Funkts. Anal. Pril., 15, No. 3, 54-66 (1981). MR0630339 (83a:58026a) 103. Yu. M. Okunev and V. A. Sadovnichii, "Model dynamical systems of one of the problems of external ballistics and their analytical solutions," In: Problems of Modern Mechanics [in Russian], Izd. Mosk. Gos. Univ., Moscow (1998), pp. 28-46. 104. Yu. M. Okunev, V. A. Privalov, and V. A. Samsonov, "Certain problems of motion of a body in a resisting medium," In: Proc. All-Union Conf. on Nonlinear Phenomena [in Russian], Nauka, Moscow (1991), pp. 140-144. 105. Yu. M. Okunev, V. A. Sadovnichii, V. A. Samsonov, and G. G. Chernyi, "A complex for modeling flight dynamics problems," Vestn. MGU, Ser. 1, Mat., Mekh., No. 6, 66-75 (1996). 106. J. Palais and S. Smale, "Theorems on structural stability," In: Sb. Per. Mat., 13, No. 2, 145-155 (1969). 107. J. Palis and W. De Melo, Geometric Theory of Dynamical Systems: An Introduction [Russian translation], Mir, Moscow (1986). MR0848132 (87m:58048) 108. A. M. Perelomov, "Several remarks on integration of equations of motion of a rigid body in an ideal fluid," Funkts. Anal. Prilozh., 15, No. 2, 83-85 (1981). MR0617480 (82j:58066) 109. I. G. Petrovskii, Lectures in the Theory of Ordinary Differential Equations [in Russian], Gostekhizdat, Moscow-Leningrad (1952). 110. V. A. Pliss, "On the roughness of differential equations assigned on a torus," Vestn. LGU, Ser. Mat., 13, 15-23 (1960). MR0126588 (23 #A3884) 111. V. A. Pliss, Nonlocal Problems of the Theory of Vibrations [in Russian], Nauka, MoscowLeningrad (1964). MR0171962 (30 #2188) 112. V. A. Pliss, Integral Sets of Periodic Systems of Differential Equations [in Russian], Nauka, Moscow (1967). 113. V. A. Pliss, "On the stability of an arbitrary system with respect to perturbations that are small in the sense of Smale," Differents. Urav. 16, No. 10, 1891-1892 (1980). MR0595574 (82b:34068) 114. V. A. Privalov and V. A. Samsonov, "On the stability of motion of an autorotating body in the flow of a medium," Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 32-38 (1990). Д 115. H. Poincare, On Curves Defined by Differential Equations [Russian translation], OGIZ, Moscow-Leningrad (1947). Д Д 116. H. Poincare, "New methods in celestial mechanics," in: H. Poincare Selected Works [Russian translation], Vols. 1, 2, Nauka, Moscow (1971, 1972). Д 117. H. Poincare, On Science [Russian translation], Nauka, Moscow (1983). MR0745803 (85k:01056) 118. R. Reissing, G. Sansonet, and R. Contie, The Qualitative Theory of Nonlinear Differential Equations [Russian translation], Nauka, Moscow (1974). MR0352601 (50 #5088) 119. S. T. Sadetov, "Conditions for the integrability of Kirchoff equations," Vestn. MGU, Ser. 1,


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140. V. V. Trofimov, "Euler's equations on finite-dimensional solvable Lie groups," Izv. Akad. Nauk SSSR. Ser. Mat., 44, No. 5, 1191-1199 (1980). MR0595263 (82e:70006) 141. E. T. Whittaker, Analytical Dynamics [Russian translation], ONTI, Moscow (1937). 142. P. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970). MR0352574 (50 #5061) 143. S. A. Chaplygin, "On the motion of massive bodies in an incompressible fluid," In: Complete Collection of Works [in Russian], Vol. 1, Izd. Akad. Nauk SSSR, Leningrad (1933), pp. 133- 135. 144. S. A. Chaplygin, Selected Works [in Russian], Nauka, Moscow (1976). MR0424503 (54 #12465) 145. F. L. Chernous'ko, L. D. Akulenko, and B. N. Sokolov, Control of Vibrations [in Russian], Nauka, Moscow (1980). 146. M. V. Shamolin, "Closed trajectories of distinct topological types in the problem of motion of a body in a resisting medium," Vestn. MGU, Ser. 1, Mat., Mekh., No. 2, 52-56, 112 (1992). MR1293705 (95d:34060) 147. M. V. Shamolin, "On the problem of motion of a body in a resisting medium," Vestn. MGU, Ser. 1, Mat., Mekh., No. 1, 52-58, 112 (1992). MR1214592 (93k:70028) 148. M. V. Shamolin, "Classification of phase portraits in the problem of motion of a body in a resisting medium under the existence of a linear damping moment," Prikl. Mat. Mekh., 57, No. 4, 40-49 (1993). MR1258007 (94i:70027) Д 149. M. V. Shamolin, "Application of Poincare map systems and reference systems in some particular systems of differential equations," Vestn. MGU, Ser. 1, Mat., Mekh., No. 2, 66-70, 113 (1993). MR1223987 (94b:34060) 150. M. V. Shamolin, "Existence and uniqueness of trajectories having infinitely remote points as limit sets for dynamical systems on the plane," Vestn. MGU, Ser. 1, Mat., Mekh., No. 1, 68-71, 112 (1993). MR1293942 (95e:34036) 151. M. V. Shamolin, "New two-parameter family of phase portraits in the problem of motion of a body in a medium," Dokl. Ross. Akad. Nauk, 337, No. 5, 611-614 (1994). MR1298329 (95g:70006) 152. M. V. Shamolin, "The definition of relative structural stability and a two-parameter family of phase portraits in the dynamics of a rigid body," Usp. Mat. Nauk, 51, No. 1, 175-176 (1996). MR1392692 (97f:70010) 153. M. V. Shamolin, "Periodic and Poisson-stable trajectories in the problem of motion of a body in a resisting medium," Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 55-63 (1996). 154. M. V. Shamolin, "A variety of types of phase portraits in the dynamics of a rigid body interacting with a resisting medium," Dokl. Ross. Akad. Nauk, 349, No. 2, 193-197 (1996). MR1440994 (98b:70009) 155. M. V. Shamolin, "An introduction to the problem on the deceleration of a body in a resisting medium and a new two-parametric family of phase portraits," Vestn. MGU, Ser. 1, Mat., Mekh., No. 4, 57-69 (1996). MR1644665 (99e:70027) 156. M. V. Shamolin, "On the integrable case in the three-dimensional dynamics of a rigid body interacting with a medium," Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 65-68 (1997).


Д 157. M. V. Shamolin, "Spatial Poincare map systems and reference systems," Usp. Mat. Nauk, 52, No. 3, 177-178 (1997). MR1479402 (99a:34089) 158. M. V. Shamolin, "On the integrability in transcendental functions," Usp. Mat. Nauk, 53, No. 3, 209-210 (1998). MR1657632 (99h:34006) 159. M. V. Shamolin, "A family of portraits with limit cycles in the plane dynamics of a rigid body interacting with a medium," Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 29-37 (1998). 160. M. V. Shamolin, "Certain classes of partial solutions in the dynamics of a rigid body interacting with a medium," Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 178-189 (1999). 161. M. V. Shamolin, "New Jacobi integrable cases in the dynamics of a rigid body interacting with a medium," Dokl. Ross. Akad. Nauk, 364, No. 5, 627-629 (1999). MR1702618 (2000k:70008) 162. M. V. Shamolin and S. V. Tsyptsin, "Analytical and numerical study of trajectories of motion of a body in a resisting medium," In: Research Report of the Institute of Mechanics of Moscow State University No. 4289 [in Russian], Moscow (1993). 163. D. Arrowsmith and C. Place, Ordinary Differential Equations. A Qualitative Approach with Applications [Russian translation], Mir, Moscow (1986). MR0684068 (85a:34001) 164. C. Jacobi, Lectures on Dynamics [in Russian], ONTI, Moscow-Leningrad (1936). 165. M. V. Jacobson, "On smooth mappings of a circle into itself," Mat. Sb., No. 85, 183-188 (1975). 166. A. Yu. Ishlinsky and D. M. Klimov, "Some aspects of the solution of the main problem of inertial navigation," J. Inst. Navig., 23, No. 4 (1970) 167. M. Peixoto, "On structural stability," Ann. Math., (2), 69, 199-222 (1959). MR0101951 (21 #753) 168. M. Peixoto, "Structural stability on two-dimensional manifolds," Topology, 1, No. 2, 101-120 (1962). MR0142859 (26 #426) 169. M. Peixoto, "On an approximation theorem of Kupka and Smale," J. Diff. Eq., 3, 214-227 (1966). MR0209602 (35 #499) 170. L. Prandtl and A. Betz, Ergebnisse der Aerodinamishen Versuchsastalt zu Gottingen, b. 4 ? ? Liefrung. Munchen-Berlin; R. Oldenbourg (1932). 171. M. V. Shamolin, "Three-dimensional structural optimization of controlled rigid motion in a resisting medium," In: Proceedings of WCSMO-2, Zakopane, Poland, May 26-30, 1997, Zakopane, Poland (1997), pp. 387-392.
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MR1930111 (2003g:70008) 70E17 Georgievski D. V. (RS-MOSC) ; Shamolin, M. V. (RS-MOSC) i, Generalized dynamic Euler equations for a rigid body with a fixed point in Rn . (Russian) Dokl. Akad. Nauk 383 (2002), no. 5, 635-637; translation in Dokl. Phys. 47 (2002), no. 4, 316-318. The authors continue the investigations they began in an earlier paper [Dokl. Akad. Nauk 380 (2001), no. 1, 47-50; MR1867984 (2003a:70002)] in which they studied the kinematics and mass geometry of an n-dimensional rigid body with a fixed point in Rn . The present paper contains a derivation of the generalized dynamic Euler equations for this problem. Using the representation of a differential equation that generalizes the classical law of the change in angular momentum of a body in terms of dual tensors, the authors obtain generalized dynamic Euler equations. They consider in detail the case when there are no external forces. For this case they show that the number of independent first integrals is less than the number of components of angular velocity by the value 1 (n - 2)(n - 1). 2 Reviewed by Gennady Victorovich Gorr c Copyright American Mathematical Society 2003, 2008

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MR1919087 (2004j:37161) 37N05 (34C05 37G15 70E15 70K50) Shamolin, M. V. Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium. Dynamical systems, 10. J. Math. Sci. (New York) 110 (2002), no. 2, 2528-2557. This paper is a translated version of the Russian original; it contains several mistakes and typos. The paper treats the problem of appearance (or disappearance) of limit cycles for a vector field on R2 . A survey is offered and old results such as the Hopf bifurcation scenario are recalled. The author shows how such bifurcations occur in a system describing the motion of a rigid body interacting with a medium. Stokes' theorem (refered to as the Gauss-Ostrogradski formula or i Д the Green formula in the present paper), as well as the Poincare-Bendixson theorem, are used repeatedly. Reviewed by Vincent Naudot References 1. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New York-Toronto, Israel Program for Scien-


2.

3. 4. 5. 6. 7. 8.

9. 10.

11. 12.

13. 14. 15. 16. 17. 18. 19.

20.

tific Translations, Jerusalem-London (1973). MR0350126 (50 #2619) A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Theory of Bifurcations of Dynamic Systems on a Plane, Halsted Press, New York-Toronto, Israel Program for Scientific Translations, Jerusalem-London (1973). MR0344606 (49 #9345) A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators, Pergamon Press, OxfordNew York-Toronto (1966). MR0198734 (33 #6888) D. V. Anosov, "Geodesic flows on closed Riemannian manifolds of negative curvature," Tr. Steklov Mat. Inst., 90, 3-209 (1967). MR0224110 (36 #7157) V. I. Arnol'd, Supplementary Chapters to the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978). MR0526218 (80i:34001) D. K. Arrowsmith and C. M. Place, Ordinary Differential Equations: A Qualitative Approach with Applications, Chapman & Hall, London-New York (1982). MR0684068 (85a:34001) N. N. Bautin, "Certain qualitative investigation methods for dynamic systems connected with field rotation," J. Appl. Math. Mech., 37, 936-941 (1974). MR0361262 (50 #13708) N. N. Bautin and E. A. Andronova-Leontovich, Methods and Rules for the Qualitative Study of Dynamical Systems on the Plane [in Russian], Mathematical Reference Library, Vol. 11, Nauka, Moscow (1990). MR1126908 (92j:34051) Д Д Д I. Bendixon, "Sur les courbes definies par des equations differentielles," Acta Math., 24, 1-88 (1901). MR1554923 A. L. Besse, Manifolds All of Whose Geodesics Are Closed, Ergebnisse der Mathematik und ihrer Grenz gebriete, Vol. 93, Springer-Verlag, Berlin-New York (1978). MR0496885 (80c:53044) G. D. Birkhoff, Dynamical Systems, Amer. Math. Soc., Colloq. Publ., Vol. 9, New York (1927). A. D. Bruno, Local Methods in Nonlinear Differential Equations. Part I. The Local method of Nonlinear Analysis of Differential Equations. Part II. The Sets of Analyticity of a Normalizing Transformation, Springer-Verlag, Berlin-New York (1989). MR0993771 (90c:58150) C. Godbillon, Geometrie Differentiable et Mecanique Analytique, Hermann, Paris (1969). Д Д MR0242081 (39 #3416) V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Heavy Rigid Body Around a Fixed Point [in Russian], Gostekhizdat, Moscow (1953). D. M. Grobman, "Topological classification of neighborhoods of a singularity in n-space," Mat. Sb., 56 (98), 77-94 (1962). MR0138829 (25 #2270) B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods and Applications [in Russian], Nauka, Moscow (1986). MR0566582 (81f:53001) V. V. Kozlov, "On a problem of a heavy rigid body falling in a resisting medium," Vestn. Mosk. Univ., Ser. Mat. Mekh., No. 1, 79-86 (1990). MR1064287 (91m:76041) S. Lefshetz, Differential Equations: Geometric Theory, Pure and Appl. Math., Vol. VI, Interscience, London (1957). MR0094488 (20 #1005) E. A. Leontovich and A. G. Maier, "On a scheme determining the topological structure of the separation of trajectories," Dokl. Akad. Nauk SSSR, 103, 557-560 (1955). MR0072305 (17,262h) E. A. Leontovich-Andronova and L. P. Shil'nikov, "The contemporary state of the theory of


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Mekh., 113, No. 2, 66-70 (1993). MR1223987 (94b:34060) 37. M. V. Shamolin, "A new two-parameter family of phase portraits in the problem of the motion of a body in a medium," Phys. Dokl., 39, No. 8, 587-590 (1994). MR1298329 (95g:70006) 38. M. V. Shamolin, "Phase portrait classification of the motion of a body in a resisting medium in the presence of a linear damping moment," J. Appl. Math. Mech., 57, No. 4, 623-632 (1993). MR1258007 (94i:70027)
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MR1914556 (2003g:34019) 34A34 (34A05 34C40 70E40) Shamolin, M. V. (RS-MOSC) On the integration of some classes of nonconservative systems. (Russian) Uspekhi Mat. Nauk 57 (2002), no. 1(342), 169-171; translation in Russian Math. Surveys 57 (2002), no. 1, 161-162. Some systems of a nonconservative type encountered in the dynamics of rigid bodies interacting with a medium are considered. Proposition. The (2n - 1)-parametric set of systems of equations on the plane R2 (x, y ),
2n-1

x = ax + by +
i=1 2n-1

i x2n-i y i x2
i=1

i-1

,

y = cx + dy +

n-(i+1) i

y,

has a (generally speaking, transcendental) first integral, expressed via elementary functions. Reviewed by L. M. Berkovich References 1. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of oscillations, Nauka, Moscow 1981; English transl., Theory of oscillators, Pergamon Press, Oxford 1966. MR0665745 (83i:34002) 2. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative theory of second-order dynamic systems, Nauka, Moscow 1966; English transl., Wiley, New York 1973. MR0350126 (50 #2619) 3. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Theory of bifurcations of dynamic systems on the plane, Nauka, Moscow 1967. (Russian) MR0235228 (38 #3539)


4. M. V. Shamolin, Uspekhi Mat. Nauk 53:3 (1998), 209-210; English transl., Russian Math. Surveys 53 (1998), 637-638. MR1657632 (99h:34006) 5. M. V. Shamolin, Dokl. Akad. Nauk 364 (1999), 627-629; English transl., Dokl. Phys. 44 (1999), 110-113. MR1702618 (2000k:70008)
Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

c Copyright American Mathematical Society 2003, 2008

Citations Article From References: 1 From Reviews: 0

MR1868040 (2002f:70005) 70E40 (70H06) Shamolin, M. V. Complete integrability of equations of motion of a spatial pendulum in an incident medium flow. (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2001, no. 5, 22-28, 70. Summary (translated from the Russian): "Previously, we considered the problem of a plane pendulum in an incident medium flow. In the present paper we construct a generalization of this problem to the spatial case. We establish the complete integrability in the sense of Jacobi of this problem. In the plane case there sometimes exists a single transcendental first integral expressed in terms of elementary functions, but in the spatial case there can be several such integrals (under certain conditions)." c Copyright American Mathematical Society 2002, 2008

Citations Article From References: 0 From Reviews: 2

MR1867984 (2003a:70002) 70B10 (70E17) Georgievski D. V. (RS-MOSC) ; Shamolin, M. V. (RS-MOSC) i, Kinematics and mass geometry of a rigid body with a fixed point in Rn . (Russian) Dokl. Akad. Nauk 380 (2001), no. 1, 47-50; translation in Dokl. Phys. 46 (2001), no. 9, 663-666. The authors consider the kinematics and mass geometry of a rigid body with a fixed point in an n-dimensional space. Using the generalized Euler formula and the angular velocity tensor they determine the velocities of the points of the body. The angular velocity tensor of (n - 2)nd rank is associated with the dual angular velocity tensor of the second rank. The authors use the generalized


Rival formula to determine the accelerations of the points of the body. In the case of hyperplane motion, they determine the components of the angular velocity tensor and the components of the dual angular velocity. They find relations for the angular momentum of the body for kinetic energy. They show that the mass geometry of an n-dimensional rigid body is determined by the second-order symmetric inertia tensor. For n = 3, the tensor I(2) , introduced to define the angular momentum, and the tensor J(2) , characterizing the kinetic energy, coincide and are the conventional inertia tensor in R3 . The results obtained are only of theoretical interest. Reviewed by Gennady Victorovich Gorr c Copyright American Mathematical Society 2003, 2008

Citations Article From References: 0 From Reviews: 0

MR1872149 (2002i:70006) 70E40 Shamolin, M. V. (RS-MOSC-MC) Integrability cases for equations of the three-dimensional dynamics of a rigid body. (Russian. English, Ukrainian summaries) Prikl. Mekh. 37 (2001), no. 6, 74-82; translation in Internat. Appl. Mech. 37 (2001), no. 6, 769-777. Summary: "A dynamic model of the interaction of a rigid body with a resisting medium under conditions of a jet flow is considered. This model allows one to extend the results for the corresponding problems from plane dynamics of a rigid body interacting with the medium and to obtain their three-dimensional analogues, as well as to establish the integrability in the sense of Jacobi of the new cases. Thus, the integrals in some cases can be expressed in terms of elementary functions. The classical problem of a spherical pendulum in a jet flow and that of the motion of a three-dimensional body with a servoconstraint are proved to be integrable. Mechanical and topological analogues of these problems are presented." c Copyright American Mathematical Society 2002, 2008

Citations Article From References: 0 From Reviews: 0


MR1844242 (2002d:90024) 90B25 Borisenok, I. T. ; Shamolin, M. V. Solution of the differential diagnostic problem by the statistical testing method. (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2001, no. 1, 29-31, 72. Summary (translated from the Russian): "The differential diagnostic problem for the functional state of control plants having a modular structure and a finite set of possible failures can be reduced to two independent sequentially solvable problems: the control problem, i.e., the establishment of a criterion for the presence of failure in the system, and the diagnostic problem, i.e., the identification of the failure. The criterion for failure in the system can be the plant trajectory leaving some prespecified surface. The failure can occur at any previously unknown instant during the motion of the plant and at any point within the specified surface. The diagnostic problem can be solved by tracking the trajectory of the plant after its departure from the control surface. We give a solution to the differential diagnostic problem for dynamical control systems in the case of trajectory measurements with noise, starting from general probabilistic considerations." c Copyright American Mathematical Society 2002, 2008

Citations Article From References: 2 From Reviews: 0

MR1833828 (2002c:70005) 70E15 Shamolin, M. V. (RS-MOSC-IMC) Integrability in the sense of Jacobi in the problem of the motion of a four-dimensional rigid body in a resisting medium. (Russian) Dokl. Akad. Nauk 375 (2000), no. 3, 343-346; translation in Dokl. Phys. 45 (2000), no. 11, 632-634. From the text (reviewer's translation): "This paper is devoted to studying the motion of a so-called four-dimensional rigid body that interacts with a resisting medium according to `streamline flows'. It is assumed that all the interactions of the rigid body with the medium are concentrated on the part of the surface of the body (three-dimensional) that has the shape of a ball (three-dimensional)." The results are as in the title. Reviewed by J. S. Joel c Copyright American Mathematical Society 2002, 2008


Citations Article From References: 1 From Reviews: 0

MR1777365 (2002d:34049) 34C05 Shamolin, M. V. (RS-MOSC) On limit sets of differential equations near singular equilibrium points. (Russian) Uspekhi Mat. Nauk 55 (2000), no. 3(333), 187-188; translation in Russian Math. Surveys 55 (2000), no. 3, 595-596. For the third-order system
2 2 = -z2 + (z1 + z2 ) sin + n2 sin cos2 + (B sin cos )/m, 0 2 z2 = n2 sin cos - z2 (, z1 , z2 ) - z1 cos / sin , 0

z1 = -z1 (, z1 , z2 ) + z1 z2 cos / sin , where
2 2 (, z1 , z2 ) = - (z1 + z2 ) cos + n2 sin2 cos - (B cos2 )/m, 0

, n0 , B , m > 0, the author proves the existence of an attracting limit cycle in the spherical layer (0,) = {(, z1 , z2 ) R3 : z1 > 0, 0 < < }. Reviewed by A. P. Sadovski i References Д 1. H. Poincare, On curves defined by differential equations, OGIZ, Moscow-Leningrad 1947 (Russian); French original in OEuvres de Henri Poincare, vol. 1, Guathier-Villars, Paris 1928. Д 2. A. A. Andronov, Collected works, Izdat. Akad. Nauk SSSR, Moscow 1956. (Russian) MR0156047 (27 #5980) 3. A. A. Andronov and E. A. Leontovich, "Some cases of the dependency of limit cycles on a parameter", Uchen. Zap. Gorkov. Gos. Univ. 1937, no. 6. (Russian) ? ? ? 4. E. Hopf, "Abzweigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems", Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Nat. Kl. 95 (1943), 3-22. MR0039141 ? (12,501c) 5. M. V. Shamolin, Uspekhi Mat. Nauk 52:3 (1997), 177-178; English transl., Russian Math. Surveys 52 (1997), 621-622. MR1479402 (99a:34089) 6. M. V. Shamolin, Dokl. Akad. Nauk 337 (1994), 611-614; English transl., Phys. Dokl. 39 (1994), 587-590. MR1298329 (95g:70006) 7. M. V. Shamolin, Dokl. Akad. Nauk 349 (1996), 193-197; English transl., Phys. Dokl. 41 (1996), 320-324. MR1440994 (98b:70009)
Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors.

c Copyright American Mathematical Society 2002, 2008


Citations Article From References: 1 From Reviews: 0

MR1776307 (2001k:70006) 70E15 (34C60 37C75 74F10) Shamolin, M. V. (RS-MOSC-MC) A new family of phase portraits in the three-dimensional dynamics of a rigid body interacting with a medium. (Russian) Dokl. Akad. Nauk 371 (2000), no. 4, 480-483; translation in Dokl. Phys. 45 (2000), no. 4, 171-174. The three-dimensional dynamics of a rigid body interacting with a viscous medium has been given an acceptable qualitative description in only the very simplest, most simplified situations. The dynamical model used in the present paper is rather simple and enables the author to form the equations of motion of the body without preliminary computations of detailed characteristics of the motion of the medium (for more details see the book by B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov [Vvedenie v zadaqu o dvi enii tela v soprotivl weis srede (Introduction to the problem of the motion of a body in a resisting medium), Izdat. Mosk. Gos. Univ., Moscow, 1986; per bibl.]). In this paper the author attempts to generalize to the threedimensional case a number of his results obtained for the plane-parallel dynamics of a body in a resisting medium. The correctness of such a generalization was discussed previously by the author [Izv. Ross. Akad. Nauk Mekh. Tverd. Tela 1997, no. 2, 65-68; RZhMat 1997:11 B291]. Thus, the author studies the fast motion of a dynamically symmetric body undergoing conditions of stream flow. The interaction of the medium with the body is concentrated at the bow (front part) of the surface of the body, which has the form of a flat disk. The author writes down the equations of motion of the body, and by excluding the cyclic variables he distinguishes an independent subsystem of three autonomous equations. Then he studies the singular points of the vector field on a three-dimensional noncompact (open) manifold. The analysis of the limit sets and the separatrices allows him to describe qualitatively the phase topology of the reduced system. Then the author introduces an index that characterizes the behavior of the separatrices of the trajectories, and using this index he classifies a countable number of topologically distinct phase portraits that occur in the given problem. Reviewed by Igor Gashenenko c Copyright American Mathematical Society 2001, 2008

Citations Article From References: 0 From Reviews: 0


MR1806854 90B25 Borisenok, I. T. ; Shamolin, M. V. (RS-MOSC) Solution of a problem of differential diagnostics. (Russian. English, Russian summaries) New computer technologies in control systems (Russian) (Pereslavl -Zalesski 1996). i, Fundam. Prikl. Mat. 5 (1999), no. 3, 775-790. {This item will not be reviewed individually. For details of the collection in which this item appears see MR1806845 (2001g:49003) .} {For the entire collection see MR1806845 (2001g:49003)} c Copyright American Mathematical Society 2008

Citations Article From References: 1 From Reviews: 0

MR1741681 (2000j:37021) 37C20 (34D30 70E99 70K99) Shamolin, M. V. (RS-MOSC) Robustness of dissipative systems and relative robustness and nonrobustness of systems with variable dissipation. (Russian) Uspekhi Mat. Nauk 54 (1999), no. 5(329), 181-182; translation in Russian Math. Surveys 54 (1999), no. 5, 1042-1043. From the text (translated from the Russian): "We present a brief survey of problems of relative structural stability (relative robustness) of dynamical systems considered not on the entire space of dynamical systems but only on some subspace of it [M. V. Shamolin, Uspekhi Mat. Nauk 51 (1996), no. 1(307), 175-176; MR1392692 (97f:70010)]." c Copyright American Mathematical Society 2000, 2008

Citations Article From References: 4 From Reviews: 0

MR1702618 (2000k:70008) 70E40 (70H06) Shamolin, M. V. (RS-MOSC-MC) New integrable, in the sense of Jacobi, cases in the dynamics of a rigid body interacting with a medium. (Russian) Dokl. Akad. Nauk 364 (1999), no. 5, 627-629; translation in Dokl. Phys. 44 (1999), no. 2, 110-113. From the text (translated from the Russian): "The dynamic model of the interaction of a rigid


body with a resisting medium under jet flow conditions that is considered not only allows us to successfully transfer the results of corresponding problems from the two-dimensional dynamics of a rigid body interacting with a medium and to obtain their three-dimensional analogues, it also reveals new Jacobi-integrable cases. Here the integrals can sometimes be expressed in terms of elementary functions. We demonstrate the integrability of the classical problem of a spherical pendulum submerged in an incident flow of a medium and the problem of the three-dimensional motion of a body in the presence of a servoconstraint. We also give mechanical and topological analogues of the latter two problems." c Copyright American Mathematical Society 2000, 2008

Citations Article From References: 3 From Reviews: 0

MR1657632 (99h:34006) 34A20 (30D30) Shamolin, M. V. (RS-MOSC) On integrability in transcendental functions. (Russian) Uspekhi Mat. Nauk 53 (1998), no. 3(321), 209-210; translation in Russian Math. Surveys 53 (1998), no. 3, 637-638. The problem of integrability of systems of ordinary differential equations in transcendental functions is discussed in this paper. Reviewed by Shamil Makhmutov c Copyright American Mathematical Society 1999, 2008

Citations Article From References: 3 From Reviews: 0

MR1479402 (99a:34089) 34C05 (34C11) Shamolin, M. V. (RS-MOSC) Д Spatial Poincare topographical systems and comparison systems. (Russian) Uspekhi Mat. Nauk 52 (1997), no. 3(315), 177-178; translation in Russian Math. Surveys 52 (1997), no. 3, 621-622. Д The notions of the Poincare topographical system, the characteristic function and the comparison system are generalised for the higher-dimensional case. Theorem. Assume that in the 1-connected domain D Rn containing a unique singular point x0 of the smooth vector field v , there exists Д the hypersurface x0 , D = such that there exists a Poincare topographical system, having a center at x0 and defined by a smooth function V , extended along up to , filling the


domain K D and such that (v , grad V )|Rn 0 in K . Then in the domain D there is no closed curve consisting of the trajectories of the vector field v and intersecting . Applications to the center-focus problem are discussed. Reviewed by Natalia Borisovna Medvedeva c Copyright American Mathematical Society 1999, 2008

Citations Article From References: 2 From Reviews: 0

MR1644665 (99e:70027) 70E99 (34C99) Shamolin, M. V. An introduction to the problem of the braking of a body in a resisting medium, and a new two-parameter family of phase portraits. (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1996, no. 4, 57-69, 112. Summary (translated from the Russian): "We actually begin with a consideration of a model version of the problem of free plane-parallel braking of a rigid body in a resisting medium under conditions of jet-type or detached flow. We carry out a qualitative analysis of the systems of differential equations that describe a given class of motions and, based on it, obtain a new twoparameter family of phase portraits consisting of an uncountable set of nonequivalent portraits without limit cycles." c Copyright American Mathematical Society 1999, 2008

Citations Article From References: 3 From Reviews: 0

MR1440994 (98b:70009) 70E15 (34C05 70K05 76B10) Shamolin, M. V. (RS-MOSC) Manifold of phase portrait types in the dynamics of a rigid body interacting with a resisting medium. (Russian) Dokl. Akad. Nauk 349 (1996), no. 2, 193-197; translation in Phys. Dokl. 41 (1996), no. 7, 320-324. The author considers a version of the plane-parallel motion of a rigid body in a resisting medium. He assumes that a part of the body's surface has the shape of a flat plate and that the interaction of the medium with the body is concentrated at precisely this part. A similar model proves useful in the investigation of bodies moving in a jet flow. By eliminating the cyclic coordinates, the equations of motion are reduced to a second-order au-


tonomous system. The author uses qualitative methods to study and classify the phase trajectories of the reduced dynamical system. In particular, he studies limit cycles and singular points of the vector field, and the behavior of stable and unstable separatrices. The presence of free parameters adds additional complexity to the system. For example, the author presents a two-parameter family of dynamical systems with a countable set of topologically different phase portraits. Reviewed by Igor Gashenenko c Copyright American Mathematical Society 1998, 2008

Citations Article From References: 2 From Reviews: 1

MR1392692 (97f:70010) 70E15 Shamolin, M. V. (RS-MOSC) Determination of relative robustness and a two-parameter family of phase portraits in the dynamics of a rigid body. (Russian) Uspekhi Mat. Nauk 51 (1996), no. 1(307), 175-176; translation in Russian Math. Surveys 51 (1996), no. 1, 165-166. The author gives a definition of the relative robustness of a system of differential equations that differs from previously used definitions. It contains two main points: sufficient smallness of the homeomorphism that produces the equivalence, and C 1 -topology in the space of vector fields. As an example, the author considers a problem that describes the dynamics of a rigid body interacting with a medium. He proves a theorem on absolute robustness, from which it follows that there exists a two-parameter family of phase portraits in which a degenerate transition occurs in the passage from one topological portrait type to another. It should be noted that the space in which the system is absolutely robust has finite measure, while the space in which the system is a system of the first degree of robustness has measure zero in the original space. Reviewed by Gennady Victorovich Gorr c Copyright American Mathematical Society 1997, 2008

Citations Article From References: 1 From Reviews: 0


MR1809236 70K99 (34D30 37C20 37N05) Shamolin, M. V. Relative structural stability of dynamical systems in the problem of the motion of a body in a medium. (Russian) Analytic, numerical and experimental methods in mechanics (Russian), 14-19, Moskov. Gos. Univ., Moscow, 1995. {This item will not be reviewed individually. For details of the collection in which this item appears see MR1809235 (2001g:00013) .} {For the entire collection see MR1809235 (2001g:00013)} c Copyright American Mathematical Society 2008

Citations Article From References: 4 From Reviews: 0

MR1298329 (95g:70006) 70E15 (34C99 70K05) Shamolin, M. V. (RS-MOSC) A new two-parameter family of phase portraits in the problem of the motion of a body in a medium. (Russian) Dokl. Akad. Nauk 337 (1994), no. 5, 611-614; translation in Phys. Dokl. 39 (1994), no. 8, 587-590. The paper deals with the Kirchhoff problem on the motion of a rigid body in an infinite ideal incompressible fluid medium. The author considers a sixth order dynamic system from which a second order subsystem splits off. The complete topological classification of phase portraits is carried out and a two-parameter family of phase portraits consisting of an uncountable set of topologically distinct phase portraits is isolated. Reviewed by V. A. Sobolev c Copyright American Mathematical Society 1995, 2008

Citations Article From References: 3 From Reviews: 0


MR1293942 (95e:34036) Shamolin, M. V. Existence and uniqueness dynamical systems on the Vestnik Moskov. Univ. Ser. I

34C35 (34C99)

of trajectories that have points at infinity as limit sets for plane. (Russian. Russian summary) Mat. Mekh. 1993, no. 1, 68-71, 112.

Summary (translated from the Russian): "We consider dynamical systems on the plane, cylinder and sphere. For some classes of systems we prove the existence and uniqueness of trajectories going out to infinity in the plane. For one-parameter systems of equations having monotonicity properties on two-dimensional oriented surfaces, we examine the problem of the existence and uniqueness of limit sets and their monotone dependence on the parameters." c Copyright American Mathematical Society 1995, 2008

Citations Article From References: 3 From Reviews: 0

MR1258007 (94i:70027) 70K99 (34C99 70K20 76B05) Shamolin, M. V. Phase portrait classification in a problem on the motion of a body in a resisting medium in the presence of a linear damping moment. (Russian. Russian summary) Prikl. Mat. Mekh. 57 (1993), no. 4, 40-49; translation in J. Appl. Math. Mech. 57 (1993), no. 4, 623-632. Summary (translated from the Russian): "We present a qualitative analysis of a dynamical system that describes a model version of the problem of the plane-parallel motion of a body in a medium with jet or separated flow when the entire interaction of the medium with the body is concentrated on a part of the surface of the body having the form of a flat plate. The force of the interaction is directed along the normal to the plate, and the point of application of this force depends only on the angle of attack. A thrust force acts along the mean perpendicular to the plate, which ensures that the value of the velocity of the center of the plate remains constant throughout the motion. In addition, a damping moment, linear with respect to the angular velocity, is imposed on the body. We carry out the phase portrait classification of the system depending on the coefficient of the moment. We note the mechanical and topological analogies with a pendulum fixed in a flowing medium." c Copyright American Mathematical Society 1994, 2008


Citations Article From References: 3 From Reviews: 0

MR1223987 (94b:34060) 34C99 (34C05 34C25 76D99) Shamolin, M. V. Д Application of the methods of Poincare topographical systems and comparison systems in some concrete systems of differential equations. (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1993, no. 2, 66-70, 113. Summary (translated from the Russian): "We consider autonomous systems on the plane or a two-dimensional cylinder and study questions of the existence for various classes of systems of Д Poincare topographical systems or more general comparative systems. As applications we consider dynamical systems that describe the plane-parallel motion of a body in a resisting medium as well as various model variants of it." c Copyright American Mathematical Society 1994, 2008

Citations Article From References: 3 From Reviews: 0

MR1293705 (95d:34060) 34C23 (34C05 34C25 70E15) Shamolin, M. V. Closed trajectories of various topological types in the problem of the motion of a body in a resisting medium. (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1992, no. 2, 52-56, 112. Summary (translated from the Russian): "We consider dynamical systems on a two-dimensional cylinder. We sharpen the theorems of Hopf, Bendixson and Dulac, after which it becomes possible to study closed trajectories of various topological types in connection with the problem of the motion of a body in a resisting medium. We give an example of a class of systems in the phase space of which there exists a continuum of closed trajectories of different types." c Copyright American Mathematical Society 1995, 2008

Citations Article From References: 3 From Reviews: 0


MR1214592 (93k:70028) 70H05 (34C05 34C25 58F40 70E15) Shamolin, M. V. On the problem of the motion of a body in a resistant medium. (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1992, no. 1, 52-58, 112. Summary (translated from the Russian): "We continue a qualitative analysis of a model variant of the interaction of a body with a resistant medium. Under the assumption that the motion is plane-parallel we completely analyze the case of constant velocity of the center of mass. We prove the presence of nonisolated periodic solutions, the absence of limit cycles and transcendental integrability, and present necessary and sufficient conditions for expressing the integral in terms of elementary functions." c Copyright American Mathematical Society 1993, 2008

Citations Article From References: 3 From Reviews: 0

MR1029730 (90k:70007) 70E99 (58F40 76D25) Samsonov, V. A. ; Shamolin, M. V. On the problem of the motion of a body in a resisting medium. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1989, no. 3, 51-54, 105. Summary (translated from the Russian): "We consider a variant of the problem on the motion of a body in a resisting medium under the assumption that the interaction of the medium with the body is confined to a part of the surface of the body, which has the form of a flat plate. For planeparallel motion we completely analyze the case of constant velocity of the center of the plate. We prove the nonexistence of auto-oscillations and prove transcendental integrability." c Copyright American Mathematical Society 1990, 2008