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Darboux Points and Integrability of Homogeneous Hamiltonian Systems with Three and More Degrees of Freedom. Nongeneric Cases
M. Przybylska Toruń Centre for Astronomy, N. Copernicus University,
Gagarina 11, PL-87–100 Toruń, Poland
Abstract:
In this paper the problem of classification of integrable natural
Hamiltonian systems with
$n$ degrees of freedom given by a Hamilton
function, which is the sum of the standard kinetic energy and a homogeneous
polynomial potential
$V$ of degree
$k>2$, is investigated. It is assumed
that the potential is not generic. Except for some particular cases a
potential
$V$ is not generic if it admits a nonzero solution of equation
$V'(\boldsymbol{d})=0$. The existence of such a solution gives very strong
integrability obstructions obtained in the frame of the Morales–Ramis
theory. This theory also gives additional integrability obstructions which
have the form of restrictions imposed on the eigenvalues
$(\lambda_1,\ldots,\lambda_n)$ of the Hessian matrix
$V''(\boldsymbol{d})$
calculated at a nonzero
$\boldsymbol{d}\in\mathbb{C}^n$ satisfying
$V'(\boldsymbol{d})=\boldsymbol{d}$. In our previous work we showed that
for generic potentials some universal relations between
$(\lambda_1,\ldots,\lambda_{n})$ calculated at various solutions of
$V'(\boldsymbol{d})=\boldsymbol{d}$ exist. These relations allow one to prove
that the number of potentials satisfying the necessary conditions for the
integrability is finite. The main aim of this paper was to show that
relations of such forms also exist for nongeneric potentials. We show
their existence and derive them for the case
$n=k=3$ applying the
multivariable residue calculus. We demonstrate the strength of the
results analyzing in details the nongeneric cases for
$n=k=3$.
Our analysis covers all the possibilities and we distinguish those cases
where known methods are too weak to decide if the potential is integrable
or not. Moreover, for
$n=k=3$, thanks to this analysis, a three-parameter
family of potentials integrable or superintegrable with additional
polynomial first integrals which seemingly can be of an arbitrarily high
degree with respect to the momenta was distinguished.
Keywords:
integrability, Hamiltonian systems, homogeneous potentials, differential Galois group.
MSC: 37J30,
70H07,
37J35,
34M35 Received: 30.05.2008
Accepted: 14.01.2009
Language: English
DOI:
10.1134/S1560354709020063
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