Lie algebras in vortex dynamics and celestial mechanics - IV: 1) Classificaton of the algebra of n vortices on a plane 2) Solvable problems of vortex dynamics 3) Algebraization and reduction in a three-body problem

Regular and Chaotic Dynamics

Volumn 4, Issue 1, 1999, Pages 23-50

  Bolsinov, A V ^a   Borisov, A V ^a   Mamaev, I S ^b  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b UDMURT STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 52649106183     PISSN: 15603547     EISSN: 14684845     Source Type: Journal    
DOI: 10.1070/rd1999v004n01abeh000097     Document Type: Article

References (28)

1. H. Aref. On the equilibrium and stability of a row of point vortices. J. Fluid Mech., v. 290, 1995, p. 167-191.
2. H. Aref. Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech., v. 15, 1983, p. 345-389.
3. H. Aref. Point vortex motions with a center of symmetry. Phys. Fluids, 1982, v. 25 (12), p. 2183-2187.
4. H. Aref, D. L. Vainstein. Point vertices exhibit asymmetric equilibria. Nature, v. 392, 1998, 23 April, p. 769-770.
5. V. I. Arnold, V. V. Kozlov, A. I. Neishtadt. Mathematical aspects of classical and celestial mechanics. 2nd ed. Encyclopedia of Mathematical Sciences, v. 3 (Dynamical systems III), New York, Springer, 1996.
6. V. I. Arnold. Mathematical methods of classical mechanics. New York, Springer, v. 60., 1984.
7. V. I. Arnold, A. B, Givental. Symplectic geometry. M., VINITI, v. 4, 1985, p. 5-140.
8. A. Barut, R. Raczka. Theory of Group Representations and Applications. PWN. Polish Scientific Publishers, 1977.
9. V. A. Bogomolov. Dynamics of the vortisity on a sphere. Izv. AN USSR, Mekh, zhid. gaza, No 6, 1977, p. 57-65.
10. A. V. Bolsinov. Compatible Poisson brackets on Lie algebras and the completeness of families of involutive functions Izv. AN USSR, ser. math., v. 55, 1991, No 1, p. 68-92.
11. A. V. Bolsinov, A. V. Borisov. Lax-representation and compatible Poisson brackets on Lie algebras. (to be published, 1999).
12. A. V. Borisov, K. V. Emelyanov. Non-integrability and stochastisity in rigid-body dynamics. Izhevsk, Izd. Udm. Univ., 1995.
13. A. V. Borisov, A. E. Pavlov. Dynamics and Statics of Vortices on a Plane and a Sphere - I. Reg. & Ch. Dynamics, v. 3, 1998, No 1, p. 28-39.
14. A. V. Borisov, V. G. Lebedev. Dynamics of Three Vortices on a Plane and a Sphere - II. General compact case. Reg. & Ch. Dynamics, v. 3, 1998, No 2, p. 99-114.
15. A. V. Borisov, V. G. Lebedev. Dynamics of three vortices on a plane and a sphere - III. Noncompact case. Problem of collapse and scattering. Reg. & Ch. Dynamics, v. 3, 1998, No 4, p. 76-90.
16. L. J. Campbell, R. M. Ziff. Vortex patterns and energies in a rotation superfluid. Phys. Rev. B., v. 20, 1979, No 5, p. 1886-1901.
17. D. N. Goryachev. On some cases of the motion of straightline vortices. Moskwa, 1898.
18. A. T. Fomenko. Symplectic geometry. M.: MGU, 1988.
19. C. G. J. Jacobi. Vorlesungen über Dynamik. Aufl. 2. Berlin: G. Reimer, 1884, 300 S.
20. E. R. van Kampen, A. Wintner. On a symmetrical canonical reduction of the problem of three bodies. Amer. J. Math., v. 59, 1937, No 1, p. 153-166.
21. G. Kirchhoff. Vorlesungen über mathematische Physik. Leipzig, 1891, 272 S.
22. J. Marsden, A. Weinstein. Reduction of Symplectic manifolds with symmetry. Rep. on Math. Phys., v. 5, 1974, No 5, p. 121-130.
23. V. V. Meleshko, M. Yu. Konstantinov. Dynamics of vortex structures. Kiev, Naukova Dumka, 1993.
24. E. A, Novikov, Yu. B. Sedov. Collapse of vortices. ZETF, v. 77, No 2(8), 1979, p. 588-597.
25. C. L. Charlier. Die Mechanik des Himmels. Walter de Gruyter & Co. 1927.
26. S. Smeil. Topology and mechanics. Uspekhi mat. nauk, v. 27, No 2, 1972, p. 77-133.
27. E. T. Whittaker. A treatise on the analytical dynamics. Ed. 3-d. Cambridge Univ. Press., 1927.
28. A. Wintner. The analytical foundation of celestial mechanics. Princeton Univ. Press, 1941.