Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases |
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Authors: | M Przybylska |
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Institution: | (1) Toruń Centre for Astronomy, N. Copernicus University, Gagarina 11, PL-87-100 Toruń, Poland |
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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′(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
(λ
1, …, λ
n
) of the Hessian matrix V″(d) calculated at a nonzero d ∈ ℂ
n
satisfying V′(d) = d. In our previous work we showed that for generic potentials some universal relations between (λ
1, …, λ
n
) calculated at various solutions of V′ (d) = 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.
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Keywords: | integrability Hamiltonian systems homogeneous potentials differential Galois group |
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