Periodic orbits of the spatial anisotropic Manev problem

Llibre, Jaume; Makhlouf, Abdenacer

doi:10.1063/1.4771902

Cited by 5 publications

(1 citation statement)

References 14 publications

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“…It was also called as the Manev potential when α = 2 and β = 1 (see [27,15,23]), and as the Schwarzschild potential when α = 3 and β = 1 (see [1]). (E4) For all 1 ≤ i < j ≤ N , F ij is given by the Lennard-Jones potential (see [21,22]), that is,…”

Section: Introductionmentioning

confidence: 99%

Existence of hyperbolic motions to a class of Hamiltonians and generalized $N$-body system via a geometric approach

Liu¹,

Yan²,

Zhou³

2021

Preprint

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For the classical N -body problem in R d with d ≥ 2, Maderna-Venturelli in their remarkable paper [Ann. Math. 2020] proved the existence of hyperbolic motions with any positive energy constant, starting from any configuration and along any noncollision configuration. Their original proof relies on the long time behavior of solutions by Chazy 1922 and Marchal-Saari 1976, on the Hölder estimate for Mañé's potential by Maderna 2012, and on the weak KAM theory.We give a new and completely different proof for the above existence of hyperbolic motions. The central idea is that, via some geometric observation, we build up uniform estimates for Euclidean length and angle of geodesics of Mañé's potential starting from a given configuration and ending at the ray along a given non-collision configuration. Note that we do not need any of the above previous studies used in Maderna-Venturelli's proof.Moreover, our geometric approach works for Hamiltonians 1 2 p 2 − F (x), where F (x) ≥ 0 is lower semicontinuous and decreases very slowly to 0 faraway from collisions. We therefore obtain the existence of hyperbolic motions to such Hamiltonians with any positive energy constant, starting from any admissible configuration and along any noncollision configuration. Consequently, for several important potentials F ∈ C 2 (Ω), we get similar existence of hyperbolic motions to the generalized N -body system ẍ = ∇ x F (x), which is an extension of Maderna-Venturelli [Ann. Math. 2020].

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Section: Introductionmentioning

confidence: 99%

Existence of hyperbolic motions to a class of Hamiltonians and generalized $N$-body system via a geometric approach

Liu¹,

Yan²,

Zhou³

2021

Preprint

View full text Add to dashboard Cite

show abstract

Existence of Hyperbolic Motions to a Class of Hamiltonians and Generalized N-Body System via a Geometric Approach

Liu

Yan

Zhou

2023

Arch Rational Mech Anal

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On the Manev spatial isosceles three-body problem

Paşca

Stoica

2019

Astrophys Space Sci

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We study the isosceles three-body problem with Manev interaction. Using a McGehee-type technique, we blow up the triple collision singularity into an invariant manifold, called the collision manifold, pasted into the phase space for all energy levels. We find that orbits tending to/ejecting from total collision are present for a large set of angular momenta. We also discover that as the angular momentum is increased, the collision manifold changes its topology.

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Periodic orbits of the spatial anisotropic Manev problem

Abstract: Abstract. We study the periodic orbits of the spatial anisotropic Manev problem which depend on three parameters.

Cited by 5 publications

References 14 publications

Existence of hyperbolic motions to a class of Hamiltonians and generalized $N$-body system via a geometric approach

Existence of hyperbolic motions to a class of Hamiltonians and generalized $N$-body system via a geometric approach

Existence of Hyperbolic Motions to a Class of Hamiltonians and Generalized N-Body System via a Geometric Approach

On the Manev spatial isosceles three-body problem

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