Geodesic
The topology of Lagrangian foliations of integrable systems with hyperelliptic Hamiltonian
Sbornik. Mathematics, Tome 202 (2011) no. 3, pp. 373-411 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the integrable Hamiltonian systems $$ (\mathbb C^2,\operatorname{Re}(dz\wedge dw),H=\operatorname{Re}f(z,w)) $$ with the additional first integral $F=\operatorname{Im}f$ which correspond to the complex Hamiltonian systems $(\mathbb C^2,dz\wedge dw,f(z,w))$ with a hyperelliptic Hamiltonian $f(z,w)=z^2+P_n(w)$, $n\in\mathbb N$. For $n\geqslant3$ the system has incomplete flows on any Lagrangian leaf $f^{-1}(a)$. The topology of the Lagrangian foliation of such systems in a small neighbourhood of any leaf $f^{-1}(a)$ is described in terms of the number $n$ and the combinatorial type of the leaf—the set of multiplicities of the critical points of the function $f$ that belong to the leaf. For odd $n$, a complex analogue of Liouville's theorem is obtained for those systems corresponding to polynomials $P_n(w)$ with simple real roots. In particular, a set of complex canonical variables analogous to action-angle variables is constructed in a small neighbourhood of the leaf $f^{-1}(0)$. Bibliography: 12 titles.
Keywords: integrable Hamiltonian system, Lagrangian foliation with singularities, leaf-wise equivalence of integrable systems, equivalence of holomorphic functions
Mots-clés : Liouville's theorem. 
@article{SM_2011_202_3_a3,
     author = {E. A. Kudryavtseva and T. A. Lepskii},
     title = {The topology of {Lagrangian} foliations of integrable systems with hyperelliptic {Hamiltonian}},
     journal = {Sbornik. Mathematics},
     pages = {373--411},
     year = {2011},
     volume = {202},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2011_202_3_a3/}
}
TY  - JOUR
AU  - E. A. Kudryavtseva
AU  - T. A. Lepskii
TI  - The topology of Lagrangian foliations of integrable systems with hyperelliptic Hamiltonian
JO  - Sbornik. Mathematics
PY  - 2011
SP  - 373
EP  - 411
VL  - 202
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2011_202_3_a3/
LA  - en
ID  - SM_2011_202_3_a3
ER  - 
%0 Journal Article
%A E. A. Kudryavtseva
%A T. A. Lepskii
%T The topology of Lagrangian foliations of integrable systems with hyperelliptic Hamiltonian
%J Sbornik. Mathematics
%D 2011
%P 373-411
%V 202
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2011_202_3_a3/
%G en
%F SM_2011_202_3_a3
E. A. Kudryavtseva; T. A. Lepskii. The topology of Lagrangian foliations of integrable systems with hyperelliptic Hamiltonian. Sbornik. Mathematics, Tome 202 (2011) no. 3, pp. 373-411. http://geodesic.mathdoc.fr/item/SM_2011_202_3_a3/

[1] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, v. 1, Chapman Hall, Boca Raton, FL, 2004 | MR | MR | Zbl | Zbl

[2] L. Gavrilov, “Abelian integrals related to Morse polynomials and perturbations of plane Hamiltonian vector fields”, Ann. Inst. Fourier (Grenoble), 49:2 (1999), 611–652 | MR | Zbl

[3] A. S. Mishchenko, A. T. Fomenko, “Generalized Liouville method of integration of Hamiltonian systems”, Funct. Anal. Appl., 12:2 (1978), 113–121 | DOI | MR | Zbl | Zbl

[4] V. V. Trofimov, A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras”, Russian Math. Surveys, 39:2 (1984), 1–67 | DOI | MR | Zbl

[5] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[6] A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Russian Math. Surveys, 44:1 (1989), 181–219 | DOI | MR | Zbl | Zbl

[7] A. T. Fomenko, Symplectic geometry, Adv. Stud. Contemp. Math., 5, Gordon and Beach, Luxembourg, 1995 | MR | MR | Zbl | Zbl

[8] R. Thom, “L'équivalence d'une fonction différentiable et d'un polynome”, Topology, 3, suppl. 2 (1965), 297–307 | DOI | MR | Zbl

[9] A. G. Khovanskii, “Newton polyhedra and the genus of complete intersections”, Funct. Anal. Appl., 12:1 (1978), 38–46 | DOI | MR | Zbl | Zbl

[10] T. A. Lepskii, “Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree”, Sb. Math., 201:10 (2010), 1511–1538 | DOI | Zbl

[11] V. A. Vasilev, Vetvyaschiesya integraly, MTsNMO, M., 2000

[12] J. Milnor, Singular points of complex hypersurfaces, Princeton Univ. Press, Princeton, NJ, 1968 | MR | MR | Zbl | Zbl