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The topology of Lagrangian foliations of integrable systems with hyperelliptic Hamiltonian
E. A. Kudryavtseva,
T. A. Lepskii M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
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, Liouville's theorem.
UDC:
517.938.5+
514.756.4
MSC: Primary
37J05; Secondary
37J35 Received: 10.06.2010 and 03.12.2010
DOI:
10.4213/sm7751
Fulltext:
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