Non-integrability of the Anisotropic Stormer Problem and the Isosceles Three-Body Problem

Νομικός, Δημήτριος; Papageorgiou, V.

doi:10.1016/j.physd.2008.10.010

Cited by 3 publications

(4 citation statements)

References 29 publications

(45 reference statements)

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“…Thus, obvious close congruence of results of the approximated solution with respect to the solution of the full ODE system can be considered as the advances of the suggested approach. Meanwhile, it should be highlighted that quasi‐periodic character of the obtained solutions is in agreement with the other results available in literature (see works among [8, 17–40] which concern the problem under consideration, where [37–40] are within the framework of the analytical approach to the study of mathematical models); besides, we should note that the elegant partial semianalytical or numerical solutions have been obtained in [23–27] for the special cases of magnetic field (with zero electric field's components).…”

Section: Discussionsupporting

confidence: 87%

Semianalytical findings for the dynamics of the charged particle in the Störmer problem

Ershkov,

Prosviryakov,

Leshchenko

et al. 2023

Math Methods in App Sciences

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In this semianalytical research, we present a new ansatz in solving the Störmer problem with numerical findings in graphical representations of solutions where dynamics of the charged particle in the classical dipole magnetic field (here, Earth's magnetic field) was investigated in detail. We have fully solved the Störmer problem for partial class of the charged particle's motion close to the equatorial plane of Earth; this result is the main important theoretical finding presented here with prescribed symmetry. Aforementioned motions, determined by Lorentz force in nonrelativistic case, thereby are explored in polar coordinates to present them in a quasi‐periodic motions in a general case (in equatorial plane of Earth). Thus, the system of momentum equations has been successfully solved numerically or semianalytically in each case and the resulting semianalytical solving algorithm can clarify the quasi‐periodic structure of such family of solutions (with reduction of their geometric symmetry around the Earth in an equatorial plane).

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Section: Discussionsupporting

confidence: 87%

Semianalytical findings for the dynamics of the charged particle in the Störmer problem

Ershkov,

Prosviryakov,

Leshchenko

et al. 2023

Math Methods in App Sciences

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show abstract

“…Bruns in [1] proved the non-existence of additional algebraic first integrals, later generalized by Julliard-Tosel [3], and more recent work like [4,5,6] prove the meromorphic non-integrability or non existence of meromorphic first integrals in some cases. All these proofs strongly suggest that the n-body problem is never integrable for n ≥ 3, even in particular cases (as proven for example for the isosceles 3-body problem in [7]). The colinear problem (in dimension 1) is a priori more difficult than the non-integrability proof of the n body problem in the plane and higher dimension, because it needs fewer additional first integrals to be integrable.…”

Section: Introductionmentioning

confidence: 78%

“…Using integrability table of [17], we see that the condition on eigenvalues (7) implies that the table A for such eigenvalues will only have zeros. So noting X 1 , .…”

Section: Second Order Variational Equationsmentioning

confidence: 97%

Integrability and Non Integrability of Some n Body Problems

Combot¹

2016

Recent Advances in Celestial and Space Mechanics

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International audienceWe prove the non integrability of the colinear 3 and 4 body problem, for any positive masses. To deal with resistant cases, we present strong integrability criterions for 3 dimensional homogeneous potentials of degree −1, and prove that such cases cannot appear in the 4 body problem. Following the same strategy, we present a simple proof of non integrability for the planar n body problem. Eventually, we present some integrable cases of the n body problem restricted to some invariant vector spaces

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“…Shibayama (2011) also showed the existence of planar periodic orbits with binary collisions for any period. See, e.g., Chiralt et al (2008), Cors et al (2009), Nomikos andPapageorgiou (2009) and Offin and Grand'maison (2005) for other recent results on the isosceles three-body problem.…”

mentioning

confidence: 99%

Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

Shibayama

Yagasaki

2011

Celest Mech Dyn Astr

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We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to "previously unknown" periodic orbits in the planar problem.

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Non-integrability of the Anisotropic Stormer Problem and the Isosceles Three-Body Problem

Cited by 3 publications

References 29 publications

Semianalytical findings for the dynamics of the charged particle in the Störmer problem

Semianalytical findings for the dynamics of the charged particle in the Störmer problem

Integrability and Non Integrability of Some n Body Problems

Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

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