2012
DOI: 10.1063/1.4746691
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Third order integrability conditions for homogeneous potentials of degree −1

Abstract: Abstract. We prove an integrability criterion of order 3 for a homogeneous potential of degree −1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V (r, θ) = r −1 h(exp(iθ)) with h ∈ C[z], deg h ≤ 3. We find then all meromorphically integrable potent… Show more

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Cited by 11 publications
(14 citation statements)
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“…So the potential V is integrable at order 2n−2 at θ = 0. At θ = π, the potential V is probably not integrable at order 3 but it seems quite difficult to prove as the eigenvalue depend on the parameter k which make the higher variational equation very complicated (this problem is analysed in [10]). We could also use the procedure presented in [20,21] where Maciejewski-Przybylska classify meromorphically homogeneous potentials of degree 3, 4, but in the case of V this will not work because this method only works for potentials without multiple Darboux points (here the Darboux point corresponding to θ = 0 is multiple for n ≥ 2).…”
Section: Examplementioning
confidence: 99%
“…So the potential V is integrable at order 2n−2 at θ = 0. At θ = π, the potential V is probably not integrable at order 3 but it seems quite difficult to prove as the eigenvalue depend on the parameter k which make the higher variational equation very complicated (this problem is analysed in [10]). We could also use the procedure presented in [20,21] where Maciejewski-Przybylska classify meromorphically homogeneous potentials of degree 3, 4, but in the case of V this will not work because this method only works for potentials without multiple Darboux points (here the Darboux point corresponding to θ = 0 is multiple for n ≥ 2).…”
Section: Examplementioning
confidence: 99%
“…Proof. We will directly use the main Theorem of [18]. After the convenient variable change which send the potential V 3 to a planar homogeneous potential with standard kinetic energy, and a rotation dilatation to put the central configuration at c = (1, 0), we find that the third order integrability condition can be written…”
Section: At Ordermentioning
confidence: 99%
“…Remark 1. One could compute the third order integrability condition for any curve (E k ), and even test if this condition could be satisfied thanks to the holonomic approach of third order variational equations in [18]. Here the restriction (m 1 , m 2 , m 3 ) ∈ R * + 3 is only for physical reasons, but a more complete study is possible.…”
Section: At Ordermentioning
confidence: 99%
“…This property was called "bounded eigenvalue property" in [7]. Then, integrable potentials in W are included in a finite union of affine spaces of type E.T h es t u d yo f higher variational equations integrability conditions for each affine space E allows us to find strong integrability conditions for potentials in W, and hopefully to prove that no unknown integrable potentials rely on these spaces E.…”
Section: Application To Parametrized Potentials 1241 Space Of Germs Of Integrable Potentialsmentioning
confidence: 99%