# Integrable Hamiltonian systems on S3 (1990)

A.T. Fomenko & Nguyen Tien Zung, Topological classification of integrable nondegenerate Hamiltonians on a constant energy three-dimensional sphere. (Russian) Uspekhi Mat. Nauk 45 (1990), no. 6(276), 91–111, 189; translation in Russian Math. Surveys 45 (1990), no. 6, 109–135

A. T. Fomenko & Nguyen Tien Zung, Topological classification of integrable nondegenerate Hamiltonians on the isoenergy three-dimensional sphere. Topological classification of integrable systems, 267–296, Adv. Soviet Math., 6, Amer. Math. Soc., Providence, RI, 1991.

(no electronic file available)

The above 2 papers, written with A.T. Fomenko in 1990 while I was his undergraduate student in Moscow, were actually just one. When you were in Russia at that time, you were allowed to publish a paper in a Russian journal, and then send essentially the same paper to an English-language special volume somewhere else !

Though this paper is quite simple, it contains a very interesting class of knots in S3, which may be called “generalized torus” or “integrable” knots. Maybe due to the beauty of its simplicity, quite a few people liked the paper and cited it.

Cited in:

1. Cordero, Alicia ; Martínez Alfaro, José ; Vindel, Pura . Bott integrable Hamiltonian systems on $S^2\times S^1$. Discrete Contin. Dyn. Syst. 22 (2008), no. 3, 587–604.
2. Campos, B.; Vindel, P. Graphs of NMS flows on $S^3$ with knotted saddle orbits and no heteroclinic trajectories Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 12, 2213–2224.
3. Robert Ghrist, On the contact topology and geometry of ideal fluids, Handbook of Mathematical Fluid Dynamics, 2007 – Elsevier.
4. Robet Ghrist, Braids and differential equations, Proceedings of the ICM 2006.
5. ! A.V. Bolsinov, A.T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman & Hall, 2004, 730 pp.
6. Ghrist, Robert ; Kin, Eiko . Flowlines transverse to knot and link fibrations. Pacific J. Math. 217 (2004), no. 1, 61–86.
7. G Goujvina, The Classification of the Bifurcations Emerging in the case of an Integrable Hamiltonian System with Two Degrees of Freedom when an Isoenergetic Surface is Non-Compact, Journal of Nonlinear Mathematical Physics, vol. 11 (2004), issue Supplement 1, p. 122.
8. Robert Ghrist & Rafal Komendarczyk, Topological features of inviscid flows, in: An introduction to geometry and topology of fluid flows, edited by Renzo Ricca, NATO Sience Series, Vol. 47, 2001.
9. ! A.V. Bolsinov, A.T. Fomenko, Integrable geodesic flows on two-dimensional surfaces, Kluwer Boston, 2000, 322 pp.
10. Campos, B. ; Martínez Alfaro, J. ; Vindel, P. Bifurcations of links of periodic orbits in non-singular Morse-Smale systems with a rotational symmetry on $S^3$. Topology Appl. 102 (2000), no. 3, 279–295.
11. Kidambi, Rangachari ; Newton, Paul K. Streamline topologies for integrable vortex motion on a sphere. Phys. D 140 (2000), no. 1-2, 95–125.
12. Etnyre, John ; Ghrist, Robert . Contact topology and hydrodynamics. I. Beltrami fields and the Seifert conjecture. Nonlinearity 13 (2000), no. 2, 441–458.
13. * Ouazzani-T. H, A.; Dekkaki, S.; Kharbach, J.; Ouazzani-Jamil, M. Bifurcation sets of the motion of a heavy rigid body around a fixed point in Goryatchev-Tchaplygin case. Nuovo Cimento Soc. Ital. Fis. B (12) 115 (2000), no. 10, 1175–1193.
14. ** J. Etnyre, R. Ghrist, Stratified integrals and unknots in inviscid flows. Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), 99–111, Contemp. Math., 246, Amer. Math. Soc., Providence, RI, 1999.
15. ** R. Ghrist, Chaotic knots and wild dynamics. Knot theory and its applications. Chaos Solitons Fractals 9 (1998), no. 4-5, 583–598.
16. ! ** R. Ghrist, Ph. Holmes, M. Sullivan, Knots and links in three-dimensional flows. Lecture Notes in Mathematics, Vol. 1654, x+208 pp., 1997.
17. ** AV Bolsinov, AT Fomenko, Application of classification theory for integrable Hamiltonian systems to geodesic flows on 2-sphere and 2-torus and to the description of the topological structure of momentum mapping near singular points, Journal of Mathematical Sciences, 78 (1996), No. 5.
18. ** V.V. Kalashnikov, On genericity of integrable Hamiltonian systems of Bott type, Math. Sbornik 81 (1995)n No. 1.
19. ** AV Bolsinov, AT Fomenko, Orbital classification of integrable Hamiltonian systems. Izvestya Mat. (1995).
20. ** AV Bolsinov, VV Kozlov, AT Fomenko, The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body, Russian Math. Surveys 50 (1995), 473–501.
21. ** K.N. Mishachev, Hamiltonian links in three-dimensional manifolds. Izv. Math. 59 (1995), no. 6, 1193–1205.
22. ** L. Gavrilov & M. Ouazzani-Jamil & R. Caboz, Bifurcation diagrams and Fomenko’s surgery on Liouville tori of the Kolossoff potential $U=rho+(1/rho)-kcosphi$, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 5, 545–564
23. ** VV Kalashnikov, Geometric description of minimax Fomenko invariants of integrable Hamiltonian systems on S3, RP3, S1 X S2, T3, Russian Math. Surveys 46 (1991), No. 4, 177.
24. ^** Nguyen Tien Zung, L. S. Polyakova, E. N. Selivanova, “Topological Classification of Integrable Geodesic Flows on Orientable Two-Dimensional Riemannian Manifolds with Additional Integral Depending on Momenta Linearly or Quadratically”, Funct. Anal. Appl., 27:3 (1993), 186–196
25. ^** Nguyen Tien Zung, “The complexity of integrable Hamiltonian systems on a prescribed three-dimensional constant-energy submanifold”, Russian Acad. Sci. Sb. Math., 75:2 (1993), 507–533