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Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree
T. A. Lepskii M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
Complex Hamiltonian systems with one degree of freedom on
$\mathbb C^2$ with the standard symplectic structure
$\omega_\mathbb C=dz\wedge dw$ and a polynomial Hamiltonian function
$f=z^2+P_n(w)$,
$n=1,2,3,4$, are studied. Two Hamiltonian systems
$(M_i,\,\operatorname{Re}\omega_{\mathbb C,i},\,H_i=\operatorname{Re}f_i)$,
$i=1,2$, are said to be Hamiltonian equivalent if there exists a complex symplectomorphism
$M_1\to M_2$ taking the vector field
$\operatorname{sgrad}H_1$ to
$\operatorname{sgrad}H_2$. Hamiltonian equivalence classes of systems
are described in the case
$n=1,2,3,4$, a completed system is defined for
$n=3,4$, and it is proved that it is Liouville integrable as a real Hamiltonian system. By restricting the real action-angle coordinates defined for the
completed system in a neighbourhood of any nonsingular leaf, real canonical coordinates are obtained for the original system.
Bibliography: 9 titles.
Keywords:
integrable Hamiltonian system, Hamiltonian equivalence of systems, incompleteness of flows of Hamiltonian fields, completed Hamiltonian system, action-angle variables.
UDC:
517.938.5+
514.756.4
MSC: Primary
37J35; Secondary
37J05,
70H06 Received: 27.02.2010 and 24.03.2010
DOI:
10.4213/sm7700
Fulltext:
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