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Symplectic topology of integrable dynamical systems. Rough topological classification of classical cases of integrability in the dynamics of a heavy rigid body. (English. Russian original) Zbl 0947.37037

J. Math. Sci., New York 94, No. 4, 1512-1557 (1999); translation from Zap. Nauchn. Semin. POMI 235, 104-183 (1996).
The main scope of this long paper is the classification of all integrable non-degenerate Hamiltonian systems of two degrees of freedom. A short review on Liouville integrable systems is presented with emphasis to the integrable cases of rigid body dynamics.
The definition of a Morse function on a smooth manifold is given. By studying the level curves of a simple Morse function on a two-dimensional surface, one can assign to the critical points of different nature, different letters with incoming or outcoming line segments, that represent schematically the nature of the bifurcation of the surface that occurs at this point. Such a representation is called an “atom”. The whole surface is reconstructed by joining its atoms along their free line segments, so that the surface is represented unambiguously by a “molecule”. A Bott function on a manifold \(Q\) is defined as a function whose critical points form non-degenerate critical submanifolds in \(Q\). It is shown that integrals of Hamiltonian systems cannot be Morse functions, but in general they are of the Bott type, and such integrals are called Bott integrals.
Two integrable systems are finely topologically equivalent on a 3-dimensional isoenergetic manifold if their Liouville foliations are diffeomorphic and the orientation of the isolated critical circles is preserved, while they are roughly topologically equivalent if their Liouville foliations can be transformed to one another by a Liouville twisting. Finely equivalent systems are also roughly equivalent but not the opposite.
The existence of an algorithm is shown, by which one can attach to a 3-dimensional isoenergetic manifold of an integrable system a molecule \(W\) and a marked molecule \(W^*\), which is actually \(W\) with some specific numbers attached on its edges. It is shown that two integrable systems are roughly topologically equivalent if their molecules \(W\) coincide, while they are finely topologically equivalent if their marked molecules \(W^*\) coincide. Since the sets \(\{ W \}\) and \(\{ W^* \}\) of all possible molecules and marked molecules are discrete, the classification of all integrable systems on their 3-dimensional isoenergetic manifolds is also discrete, a fact that was not obvious beforehand.
The complexity of a molecule is defined as given by the integer \(m\) (the total number of vertices) and \(n\) (the total number of edges of the molecule). Let \(\lambda (m,n)\) denote the number of molecules with complexity \((m,n)\). This number, which is zero for \(n>[3m/2]\), is finite and can be calculated for small complexities, but grows rapidly with \(m\). The interesting fact that all integrable systems with “physical interest” lie on the plane \((m,n)\) in a narrow zone along the line \(n=m-1\) is pointed out. A realization of the simple atoms that appear in a molecule is presented.
After a short historical commentary, the cases of geodesic flows on the 2-sphere and 2-torus are reviewed and a topological classification of the classical cases of integrability in the rigid body dynamics is presented.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology

References:

[1] A. T. Fomenko, ”Qualitative geometric theory of integrable systems. Classification of isoenergy surfaces and bifurcation of Liouville tori at the critical energy values,”Lect. Notes Math.,1334, 221–245 (1988). · doi:10.1007/BFb0080431
[2] S. V. Matveev and A. T. Fomenko, ”Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds,”Usp. Mat. Nauk,43, No. 1, 5–22 (1988). · Zbl 0671.58008
[3] A. T. Fomenko and H. Zieschang, ”On typical topological properties of integrable Hamiltonian systems,”Izv. AN SSSR,52, No. 2, 378–407 (1988). · Zbl 0647.58018
[4] A. T. Fomenko, ”Topological invariants of Hamiltonian systems completely integrable in the sense of Liouville,”Func. Anal. Appl.,22, No. 4, 38–51 (1988).
[5] A. T. Fomenko, ”Integrability and nonintegrability in geometry and mechanics. (Monograph),”Kluwer Acad. Publ., 1–343 (1988). · Zbl 0675.58018
[6] A. V. Bolsinov, S. V. Matveev, and A. T. Fomenko, ”Topological classification of integrable Hamiltonian systems with two degrees of freedom. The list of all systems with low complexity,”Usp. Mat. Nauk,45, No. 2, 49–77 (1990). · Zbl 0696.58019
[7] A. T. Fomenko and H. Zieschang, ”Topological invariant and criteria of equivalence of integrable Hamiltonian systems with two degrees of freedom,”Izv. AN SSSR,54, No. 3, 546–575 (1990). · Zbl 0705.58023
[8] A. T. Fomenko and T. Z. Nguen, ”Topological classification of integrable Hamiltonians on isoenergy three-dimensional sphere,”Usp. Mat. Nauk,45, No. 6 91–111 (1990).
[9] A. T. Fomenko, ”Topological classification of all Hamiltonian differential equations of general type with two degrees of freedom,” in:The Geometry of Hamiltonian Systems. Proceedings of a Workshop Held June 5–16, 1989, Springer-Verlag, New York (1991), pp. 131–339.
[10] A. A. Oshemkov, ”Topology of isoenergy surfaces and bifurcation diagrams of integrable cases of dynamic of the rigid body on so (4),”Usp. Mat. Nauk,42, No. 6, 199–200 (1987). · Zbl 0648.58016
[11] A. T. Fomenko, ”Topological classification of integrable Hamiltonian systems,”Am. Math. Soc. Adv. Sov. Math.,6, 297–306 (1991).
[12] T. Z. Nguen, ”On the property of general position of simple Bott integrals,”Usp. Mat. Nauk.,45, No. 4 161–162 (1990).
[13] A. A. Oshemkov, ”Fomenko invariants for the mail integrable cases of the rigid body motion equations,” in:Topological Classification of Integrable Hamiltonian Systems. Am. Math. Soc. Adv. Sov. Math., vol. 6, 1991, pp. 67–146. · Zbl 0745.58028
[14] A. V. Bolsinov, ”Methods for calculation of the Fomenko-Zieschang invariant,” in:Topological Classification of Integrable Hamiltonian Systems. Am. Math. Soc. Adv. Sov. Math., vol. 6, 1991, pp. 147–184. · Zbl 0744.58029
[15] E. N. Selivanova, ”Topological classification of integrable Bott geodesic flows on two-dimensional torus,” in:Topological Classification of Integrable Hamiltonian Systems. Am. Math. Soc. Adv. Sov. Math., vol. 6, 1991, pp. 209–228. · Zbl 0744.58034
[16] T. Z. Nguen, ”On the complexity of integrable Hamiltonian systems on three-dimensional isoenergy submanifolds,”Topological Classification of Integrable Hamiltonian Systems Am. Math. Soc. Adv. Sov. Math.,6, 229–256 (1991).
[17] V. V. Kalashnikov (Jr.), ”Description of the structure of Fomenko invariant on the boundary and insideQ-domains. Estimates of their number on the lower boundary for the manifoldsS 3,RP 3,S 1{\(\times\)}S 2, andT 2,”Topological Classification of Integrable Hamiltonian Systems, Am. Math. Soc. Adv. Sov. Math.,6, 297–306 (1991).
[18] S. B. Katok, ”Bifurcation sets and integral manifolds in the problem of rigid body motion,”Usp. Mat. Nauk,27, No. 2 126–132 (1972).
[19] M. P. Kharlamov,Topological Analysis of Classical Integrable Systems in Rigid Body Dynamics, Leningrad Univ. Press. (1988).
[20] A. Iacob, ”Invariant manifolds in the motion of a rigid body about a fixed point,”Rev. Roum. Math. Pur. Appl.,16, No. 10 (1971). · Zbl 0227.70003
[21] R. Cushman and H. Knorrer, ”The energy momentum mapping of the Lagrange top,”Lect. Notes Math.,1139 (1984).
[22] Ja. V. Tatarinov, ”The portraits of classical integrals in the problem of a rigid body motion about the fixed point,”Vest. MGU.,6, 99–105 (1974). · Zbl 0291.70003
[23] T. Z. Nguen and L. S. Polyakova, ”A topological classification of integrable geodesic flows on the twodimensional sphere with quadratic in momenta additional integral,”J. Nonl. Sci.,6 (1992).
[24] N. M. Ercolani and D. M. McLaughlin, ”Toward a topological classification of integrable PRE’s,” in:Geometry of Hamiltonian Systems. Proc. of a Workshop, 1989, Berkeley, MSRI. Springer-Verlag (1991).
[25] I. Stewart, ”Lowering the volume,”Nature,338, 375–376 (1980). · doi:10.1038/338375a0
[26] M. L. Byalyi, ”On polynomial in momenta first integrals for mechanical systems on two-dimensional torus,”Func. Anal. Its. Appl.,21, No. 4, 64–65 (1987). · Zbl 0626.43003 · doi:10.1007/BF01077990
[27] V. V. Kozlov and D. V. Tretschev, ”On integrability of Hamiltonian systems with torical configuration space,”Mat. Sb.,135 (177), No. 1, 119–138 (1988).
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