Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem

Fenucci, Marco; Jorba, Àngel

doi:10.1016/j.cnsns.2019.105105

Cited by 6 publications

(3 citation statements)

References 26 publications

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“…From the numerical point of view, the figure-eight solution is computed in two steps (see e.g. [11,12] for details):…”

Section: The Figure-eight Solution Of the 3-body Problemmentioning

confidence: 99%

Local minimality properties of circular motions in $1/r^α$ potentials and of the figure-eight solution of the 3-body problem

Fenucci¹

2022

Preprint

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We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local minimizer. We then apply the theory to circular orbits of the Kepler problem with potentials of type 1/r α , α > 0. By using numerical computations, we show that circular solutions are strong local minimizers for α > 1, while they are saddle points for α ∈ (0, 1). Moreover, we show that for α ∈ (1, 2) the global minimizer of the action over periodic curves with degree 2 with respect to the origin could be achieved on non-collision and non-circular solutions. After, we take into account the figure-eight solution of the 3-body problem, and we show that it is a strong local minimizer over a particular set of symmetric periodic loops.

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“…From the numerical point of view, the figure-eight solution is computed in two steps (see e.g. [11,12] for details):…”

Section: The Figure-eight Solution Of the 3-body Problemmentioning

confidence: 99%

Local minimality properties of circular motions in $1/r^α$ potentials and of the figure-eight solution of the 3-body problem

Fenucci¹

2022

Preprint

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“…This problem is referred to as the charged (n + 1)-body problem [8,1], the n-electron atom problem [4] and the Coulomb (n + 1)-body problem [6] (and the references therein). An interesting feature of this problem is the existence of spatial relative equilibria, in contrast to the n-body problem where all relative equilibria must be planar [1].…”

Section: Introductionmentioning

confidence: 99%

Spatial relative equilibria and periodic solutions of the Coulomb $(n+1)$-body problem

Constantineau¹,

García-Azpeitia²,

Lessard³

2021

Preprint

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We study a classical model for the atom that considers the movement of n charged particles of charge −1 (electrons) interacting with a fixed nucleus of charge µ > 0. We show that two global branches of spatial relative equilibria bifurcate from the n-polygonal relative equilibrium for each critical values µ = s k for k ∈ [2, ..., n/2]. In these solutions, the n charges form n/hgroups of regular h-polygons in space, where h is the greatest common divisor of k and n. Furthermore, each spatial relative equilibrium has a global branch of relative periodic solutions for each normal frequency satisfying some nonresonant condition. We obtain computer-assisted proofs of existence of several spatial relative equilibria on global branches away from the npolygonal relative equilibrium. Moreover, the nonresonant condition of the normal frequencies for some spatial relative equilibria is verified rigorously using computer-assisted proofs.

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“…We shall show some examples, with different symmetry constraints. In particular, we shall consider symmetries defined by the group Z 4 (leading to the Hip-Hop solution [10]), by Z 2 × Z 2 , and by the rotation groups of Platonic polyhedra (used for instance in [18,[20][21][22]24]). Γ-convergence was already applied to the N -body problem in [21], where the authors considered the exponent α of the potential as a parameter, and studied the behavior of the minimizers as α → +∞.…”

Section: Introductionmentioning

confidence: 99%

Symmetric constellations of satellites moving around a central body of large mass

Fenucci¹,

Gronchi²

2020

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We consider a (1 + N )-body problem in which one particle has mass m0 ≫ 1 and the remaining N have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the N bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form U ∼ 1 r α , where α ∈ [1, 2) and r is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use Γ-convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional Γ-converges to the action functional of a Kepler problem, defined on a suitable set of loops. Minimizers of the Γ-limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of m0. We discuss some examples, where the symmetry is defined by an action of the groups Z4 , Z2 × Z2 and the rotation groups of Platonic polyhedra on the set of loops.

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Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem

Cited by 6 publications

References 26 publications

Local minimality properties of circular motions in $1/r^α$ potentials and of the figure-eight solution of the 3-body problem

Local minimality properties of circular motions in $1/r^α$ potentials and of the figure-eight solution of the 3-body problem

Spatial relative equilibria and periodic solutions of the Coulomb $(n+1)$-body problem

Symmetric constellations of satellites moving around a central body of large mass

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