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Spatial relative equilibria and periodic solutions of the Coulomb $(n+1)$-body problem
Dynamical Systems · Submitted 11 Jul 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that two global branches of spatial relative equilibria bifurcate from the $n$-polygonal relative equilibrium for each critical values $\mu =s_{k}$ for $k\in \lbrack 2,...,n/2]$.
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The abstract, sentence by sentence
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 $\mu >0$. We show that two global branches of spatial relative equilibria bifurcate from the $n$-polygonal relative equilibrium for each critical values $\mu =s_{k}$ for $k\in \lbrack 2,...,n/2]$. In these solutions, the $n$ charges form $n/h$-groups 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 the existence of several spatial relative equilibria on global branches away from the $n$-polygonal relative equilibrium. Moreover, the nonresonant condition of the normal frequencies for some spatial relative equilibria is verified rigorously using computer-assisted proofs.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $\mu >0$. | background 0.91 | 0.21 | 0.03 | asserted 0.95 |
| s1 | We show that two global branches of spatial relative equilibria bifurcate from the $n$-polygonal relative equilibrium for each critical values $\mu =s_{k}$ for $k\in \lbrack 2,...,n/2]$. | result 0.98 | 0.96 | 0.45 | asserted 1.00 |
| s2 | In these solutions, the $n$ charges form $n/h$-groups of regular $h$-polygons in space, where $h$ is the greatest common divisor of $k$ and $n$. | result 0.97 | 0.83 | 0.13 | asserted 1.00 |
| s3 | Furthermore, each spatial relative equilibrium has a global branch of relative periodic solutions for each normal frequency satisfying some nonresonant condition. | result 0.98 | 0.91 | 0.20 | asserted 1.00 |
| s4 | We obtain computer-assisted proofs of the existence of several spatial relative equilibria on global branches away from the $n$-polygonal relative equilibrium. | result 0.58 | 0.97 | 0.17 | asserted 1.00 |
| s5 | Moreover, the nonresonant condition of the normal frequencies for some spatial relative equilibria is verified rigorously using computer-assisted proofs. | contribution 0.43 | 0.95 | 0.09 | asserted 1.00 |
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Original abstract
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 $\mu >0$. We show that two global branches of spatial relative equilibria bifurcate from the $n$-polygonal relative equilibrium for each critical values $\mu =s_{k}$ for $k\in \lbrack 2,...,n/2]$. In these solutions, the $n$ charges form $n/h$-groups 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 the existence of several spatial relative equilibria on global branches away from the $n$-polygonal 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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