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Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability

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Abstract

This paper is concerned with the problem of three vortices on a sphere S2 and the Lobachevsky plane L2. After reduction, the problem reduces in both cases to investigating a Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to study it using the methods of Poisson geometry. This paper presents a topological classification of types of symplectic leaves depending on the values of Casimir functions and system parameters.

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References

  1. Aref, H., Motion of Three Vortices, Phys. Fluids, 1988, vol. 31. no. 6. pp. 1392–1409.

    Article  MathSciNet  Google Scholar 

  2. Aref, H. and Stremler, M.A., Four-Vortex Motion with Zero Total Circulation and Impulse, Phys. Fluids, 1999, vol. 11, no. 12, pp. 3704–3715.

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnol’d, V. I., Kozlov, V.V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.

    Google Scholar 

  4. Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Three Vortex Sources, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 694–701.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Vortex Sources in a Deformation Flow, Regul. Chaotic Dyn., 2016, vol. 21, no. 3, pp. 367–376.

    Article  MathSciNet  MATH  Google Scholar 

  6. Boatto, S. and Koiller, J., Vortices on Closed Surfaces, in Geometry, Mechanics, and Dynamics, D. E. Chang, D.D. Holm, G. Patrick, T. Ratiu (Eds.), Fields Inst. Commun., vol. 73, New York: Springer, 2015, pp. 185–237.

    MathSciNet  MATH  Google Scholar 

  7. Bogomolov, V.A., On Two-Dimensional Hydrodynamics of a Sphere, Izv. Atmos. Ocean. Phys., 1979, vol. 15, no. 1, pp. 18–22; see also: Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana, 1979, vol. 15, no. 1, pp. 29–35.

    Google Scholar 

  8. Bogomolov, V.A., Interaction of Vortices in Plane-Parallel Flow, Izv. Akad. Nauk SSSR. Fiz. Atmos. Okeana, 1981, vol. 17, no. 2, pp. 199–201 (Russian).

    Google Scholar 

  9. Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Lie Algebras in Vortex Dynamics and Celestial Mechanics: 4, Regul. Chaotic Dyn., 1999, vol. 4, no. 1, pp. 23–50.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318; see also: Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71–132.

    Article  MathSciNet  MATH  Google Scholar 

  11. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Reduction and Chaotic Behavior of Point Vortices on a Plane and a Sphere, Discrete Contin. Dyn. Syst., 2005, suppl., pp. 100–109.

    Google Scholar 

  12. Borisov, A.V., Kilin, A.A., Mamaev, I. S., The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem, Regul. Chaotic Dyn., 2013, vol. 18, no. 1–2, pp. 33–62.

    Article  MATH  Google Scholar 

  13. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., A New Integrable Problem of Motion of Point Vortices on the Sphere, in Proc. of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, 25–30 August, 2006), A.V. Borisov, V.V. Kozlov, I. S. Mamaev, and M.A. Sokolovisky (Eds.), Dordrecht: Springer, 2008, pp. 39–53.

    Chapter  Google Scholar 

  14. Borisov, A.V., Kilin, A.A., Mamaev, I. S., Tenenev, V.A., The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid Fluid Dynam. Res., 2014, vol. 46, no. 3, 031415, 16 pp.

    Article  MathSciNet  MATH  Google Scholar 

  15. Borisov, A. V. and Lebedev, V.G., Dynamics of Three Vortices on a Plane and a Sphere: 2. General Compact Case, Regul. Chaotic Dyn., 1998, vol. 3. no. 2. pp. 99–114.

    Article  MathSciNet  MATH  Google Scholar 

  16. Borisov, A. V. and Lebedev, V.G., Dynamics of Three Vortices on a Plane and a Sphere: 3. Noncompact Case. Problems of Collapse and Scattering. Regul. Chaotic Dyn., 1998, vol. 3, no. 4, pp. 74–86.

    Article  MATH  Google Scholar 

  17. Borisov, A.V. and Mamaev, I. S., Mathematical Methods in the Dynamics of Vortex Structures, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).

    MATH  Google Scholar 

  18. Borisov, A.V. and Mamaev, I. S., On the Problem of Motion Vortex Sources on a Plane, Regul. Chaotic Dyn., 2006, vol. 11, no. 4, pp. 455–466.

    Article  MathSciNet  MATH  Google Scholar 

  19. Borisov A.V., Mamaev I. S. Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Izhevsk: R&C Dynamics, 1999 (Russian).

    MATH  Google Scholar 

  20. Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Reduction and Chaotic Behavior of Point Vortices on a Plane and a Sphere, Nelin. Dinam., 2005, vol. 1, no. 2, pp. 233–246 (Russian).

    Article  MATH  Google Scholar 

  21. Borisov, A. V. and Pavlov, A.E., Dynamics and Statics of Vortices on a Plane and a Sphere, Regul. Chaotic Dyn., 1998, vol. 3, no. 1, pp. 28–38.

    Article  MathSciNet  MATH  Google Scholar 

  22. Cari˜nena, J. F., Ra˜nada, M. F., and Santander, M., The Kepler Problem and the Laplace–Runge–Lenz Vector on Spaces of Constant Curvature and Arbitrary Signature, Qual. Theory Dyn. Syst., 2008, vol. 7, no. 1, pp. 87–99.

    Article  MathSciNet  Google Scholar 

  23. Castilho, C. and Machado, H., The N-Vortex Problem on a Symmetric Ellipsoid: A Perturbation Approach, J. Math. Phys., 2008, vol. 49, no. 2, 022703, 12 pp.

    Article  MathSciNet  MATH  Google Scholar 

  24. Dufour, J.P. and Haraki, A., Rotationnels et structures de Poisson quadratiques, C. R. Acad. Sci. Paris Sér. 1. Math., 1991, vol. 312, no. 1, pp. 137–140.

    MathSciNet  MATH  Google Scholar 

  25. Fridman, A.A. and Polubarinova, P.Ya., On Moving Singularities of a Flat Motion of an Incompressible Fluid, Geofiz. Sb., 1928, vol. 5, no. 2, pp. 9–23 (Russian).

    Google Scholar 

  26. Gonchar, V.Yu., Ostapchuk, P. N., Tur, A.V., and Yanovsky, V. V., Dynamics and Stochasticity in a Reversible System Describing Interaction of Point Vortices with a Potential Wave, Phys. Lett. A, 1991, vol. 152, no. 5–6, pp. 287–292.

    Article  MathSciNet  Google Scholar 

  27. Goldman, R., Curvature Formulas for Implicit Curves and Surfaces, Comput. Aided Geom. Design, 2005, vol. 22, no. 7, pp. 632–658.

    Article  MathSciNet  MATH  Google Scholar 

  28. Greenhill, A. G., Plane Vortex Motion, Quart. J. Pure Appl. Math., 1877/78, vol. 15, no. 58, pp. 10–27.

    MATH  Google Scholar 

  29. Gryanik, V.M., Sokolovskiy, M.A., and Verron, J., Dynamics of Heton-Like Vortices, Regul. Chaotic Dyn., 2006, vol. 11, no. 3, pp. 383–434.

    Article  MathSciNet  MATH  Google Scholar 

  30. Hwang, S. and Kim, S.-Ch., Relative Equilibria of Point Vortices on the Hyperbolic Sphere, J. Math. Phys., 2013, vol. 54, no. 6, 063101, 15 pp.

    Article  MathSciNet  MATH  Google Scholar 

  31. Hwang, S. and Kim, S.-Ch., Point Vortices on Hyperbolic Sphere, J. Geom. Phys., 2009, vol. 59, no. 4, pp. 475–488.

    Article  MathSciNet  MATH  Google Scholar 

  32. Kidambi, R. and Newton, P.K., Collision of Three Vortices on a Sphere, Il Nuovo Cimento C, 1999, vol. 22, no. 6, pp. 779–792.

    Google Scholar 

  33. Kidambi, R. and Newton, P.K., Motion of Three Point Vortices on a Sphere, Phys. D, 1998, vol. 116, nos. 1–2, pp. 143–175.

    Article  MathSciNet  MATH  Google Scholar 

  34. Kimura, Y., Vortex Motion on Surfaces with Constant Curvature, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 1999, vol. 455, no. 1981, pp. 245–259.

    Article  MathSciNet  MATH  Google Scholar 

  35. Kilin, A.A., Borisov, A. V., and Mamaev, I. S., The Dynamics of Point Vortices Inside and Outside a Circular Domain, in Basic and Applied Problems of the Theory of Vortices, A. V. Borisov, I. S. Mamaev, M.A. Sokolovskiy (Eds.), Izhevsk: R&C Dynamics, Institute of Computer Science, 2003, pp. 414–440 (Russian).

    Google Scholar 

  36. Kobayashi, Sh., Transformation Groups in Differential Geometry, Berlin: Springer, 1995.

    MATH  Google Scholar 

  37. Kozlov, V.V., Dynamical Systems 10: General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003.

  38. Koshel, K.V. and Ryzhov, E.A., Parametric Resonance with a Point-Vortex Pair in a Nonstationary Deformation Flow, Phys. Lett. A, 2012, vol. 376, no. 5, pp. 744–747.

    Article  MATH  Google Scholar 

  39. Lamb, H., Hydrodynamics, 6th ed., New York: Dover, 1945.

    MATH  Google Scholar 

  40. Middelkamp, S., Kevrekidis, P. G., Frantzeskakis, D. J., Carretero-González, R., and Schmelcher, P., Bifurcations, Stability, and Dynamics of Multiple Matter-Wave Vortex States, Phys. Rev. A, 2010, vol. 82, no. 1, 013646, 15 pp.

    Google Scholar 

  41. Montaldi, J. and Nava-Gaxiola, C., Point Vortices on the Hyperbolic Plane, J. Math. Phys., 2014, vol. 55, no. 10, 102702, 14 pp.

    Article  MathSciNet  MATH  Google Scholar 

  42. Murray, A.V., Groszek, A. J., Kuopanportti, P., and Simula, T., Hamiltonian Dynamics of Two Same-Sign Point Vortices, Phys. Rev. A, 2016, vol. 93, no. 3, 033649, 8 pp.

    Article  Google Scholar 

  43. Newton, P.K., The N-Vortex Problem: Analytical Techniques, Appl. Math. Sci., vol. 145, New York: Springer, 2001.

  44. Ragazzo, C. and Viglioni, H., Hydrodynamic Vortex on Surfaces, J. Nonlinear Sci., 2017, vol. 27, no. 5, pp. 1609–1640.

    Article  MathSciNet  MATH  Google Scholar 

  45. Ohsawa, T., Symplectic Reduction and the Lie–Poisson Dynamics of N Point Vortices on the Plane, arXiv:1808.01769 (2018).

    Google Scholar 

  46. Patera, J., Sharp, R.T., and Winternitz, P., Invariants of Real Low Dimension Lie Algebras, J. Math. Phys., 1976, vol. 17, no. 6, pp. 986–994.

    Article  MathSciNet  MATH  Google Scholar 

  47. Ryzhov, E.A. and Koshel, K.V., Dynamics of a Vortex Pair Interacting with a Fixed Point Vortex, Europhys. Lett., 2013, vol. 102, no. 4, 44004, 6 pp.

    Article  Google Scholar 

  48. Sakajo, T., Integrable four-vortex motion on sphere with zero moment of vorticity, Phys. Fluids, 2007, vol. 19, no. 1, 017109, 10 pp.

    Article  MathSciNet  MATH  Google Scholar 

  49. Sokolov, S.V. and Ryabov, P. E., Bifurcation Analysis of the Dynamics of Two Vortices in a Bose–Einstein Condensate. The Case of Intensities of Opposite Signs, Regul. Chaotic Dyn., 2017, vol. 22, no. 8, pp. 976–995.

    Article  MathSciNet  Google Scholar 

  50. Sokolovskii, M. A. and Verron, J., New Stationary Solutions to the Problem of Three Vortices in a Two-Layer Fluid, Dokl. Phys., 2002, vol. 47, no. 3, pp. 233–237; see also: Dokl. Akad. Nauk, 2002, vol. 383, no. 1, pp. 61–66.

    Article  MathSciNet  Google Scholar 

  51. Sokolovskiy, M. A. and Verron, J., Dynamics of Three Vortices in a Two-Layer Rotating Fluid, Regul. Chaotic Dyn., 2004, vol. 9, no. 4, pp. 417–438.

    Article  MathSciNet  MATH  Google Scholar 

  52. Stewart, H. J., Periodic Properties of the Semi-Permanent Atmospheric Pressure Systems, Quart. Appl. Math., 1943, vol. 1, no. 3, pp. 262–267.

    Article  MathSciNet  MATH  Google Scholar 

  53. Synge, J. L., On the Motion of Three Vortices, Canadian J. Math., 1949, vol. 1, pp. 257–270.

    Article  MathSciNet  MATH  Google Scholar 

  54. Tavantzis, J. and Ting, L., The Dynamics of Three Vortices Revisited, Phys. Fluids, 1988, vol. 31, no. 6, pp. 1392–1409.

    Article  MathSciNet  MATH  Google Scholar 

  55. Timorin, V., Topological Regluing of Rational Functions, Invent. Math., 2010, vol. 179, no. 3, pp. 461–506.

    Article  MathSciNet  MATH  Google Scholar 

  56. Vaisman, I., Lectures on the Geometry of Poisson Manifolds, Progr. Math., vol. 118, Boston,Mass.: Birkhäuser, 2012.

  57. Vetchanin, E.V. and Kazakov, A.O., Bifurcations and Chaos in the Dynamics of Two Point Vortices in an Acoustic Wave, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2016, vol. 26, no. 4, 1650063, 13 pp.

    Article  MathSciNet  MATH  Google Scholar 

  58. Vetchanin, E.V. and Mamaev, I. S., Dynamics of Two Point Vortices in an External Compressible Shear Flow, Regul. Chaotic Dyn., 2017, vol. 22, no. 8, pp. 893–908.

    Article  MathSciNet  Google Scholar 

  59. Volterra, V., Leçons sur la théorie mathématique de la lutte pour la vie, Paris: Gauthier-Villars, 1931.

    MATH  Google Scholar 

  60. Zermelo, E. F. F., Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche, in Gesammelte Werke: Vol. 2, H.-D. Ebbinghaus, C.G. Fraser, A. Kanamori (Eds.), Schriften der Mathematisch-naturwissenschaftlichen Klasse, vol. 23, Berlin: Springer, 2013, pp. 300–463.

    Google Scholar 

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Authors and Affiliations

  1. Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    Alexey V. Borisov & Ivan A. Bizyaev

  2. Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700, Russia

    Ivan S. Mamaev

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  1. Alexey V. Borisov
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  2. Ivan S. Mamaev
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  3. Ivan A. Bizyaev
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Correspondence to Alexey V. Borisov.

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Borisov, A.V., Mamaev, I.S. & Bizyaev, I.A. Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability. Regul. Chaot. Dyn. 23, 613–636 (2018). https://doi.org/10.1134/S1560354718050106

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