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Computing necessary integrability conditions for planar parametrized homogeneous potentials
TLDR
We design an algorithm that computes polynomial conditions on the parameters of a rationally parametrized planar homogeneous potential such that the dynamical system associated to the potential <i>V</i> is integrable. Expand
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A Binomial-like Matrix Equation
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We show that a pair of matrices satisfying a certain algebraic identity, reminiscent of the binomial theorem, must have the same characteristic polynomial. Expand
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A note on algebraic potentials and Morales–Ramis theory
We present various properties of algebraic potentials, and then prove that some Morales–Ramis theorems readily apply for such potentials even if they are not in general meromorphic potentials. ThisExpand
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Third order integrability conditions for homogeneous potentials of degree -1
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 exceptExpand
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Symbolic Computations of First Integrals for Polynomial Vector Fields
TLDR
In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J Pereira. Expand
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Integrability conditions at order 2 for homogeneous potentials of degree -1
We prove a meromorphic integrability condition at order 2 near a homothetic orbit for a meromorphic homogeneous potential of degree -1, which extend the Morales Ramis conditions of order 1.Expand
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Meromorphically integrable homogeneous potentials with multiple Darboux points
We prove that the only meromorphically integrable planar homogeneous potential of degree k -2,0,2 having a multiple Darboux point is the potential invariant by rotation. This case is a singular caseExpand
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Integrability conditions of geodesic flow on homogeneous Monge manifolds
Abstract We prove a meromorphic integrability criterion for the geodesic flow of an algebraic manifold of the form ${z}^{p} - f({x}_{1} , \ldots , {x}_{n} )= 0$ with the induced metric of ${Expand
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Integrable homogeneous potentials of degree −1 in the plane with small eigenvalues
We give a complete classification of meromorphically integrable homogeneous potentials V of degree −1 which are real analytic on R 2 \ {0}. In the more general case when V is only meromorphic on anExpand
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Bi-homogeneity and integrability of rational potentials
Abstract In this paper we consider natural Hamiltonian systems with two degrees of freedom for which Hamiltonian function has the form H = 1 2 ( p 1 2 + p 2 2 ) + V ( q 1 , q 2 ) and potential V ( qExpand
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