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Partially rigid motions in the n-body problem
Dynamical Systems · Submitted 25 Jul 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The goal of this paper is to show that hinged solutions don't exist when n=3 or n=4.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
A solution of the n-body problem in R^d is a relative equilibrium if all of the mutual distance between the bodies are constant. In other words, the bodies undergo a rigid motion. Here we investigate the possibility of partially rigid motions, where some but not all of the distances are constant. In particular, a {\em hinged} solution is one such that exactly one mutual distance varies. The goal of this paper is to show that hinged solutions don't exist when n=3 or n=4. For n=3 this means that if 2 of the 3 distances are constant so is the third and for n=4, if 5 of the 6 distances are constant, so is the sixth. These results hold independent of the dimension d of the ambient space.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A solution of the n-body problem in R^d is a relative equilibrium if all of the mutual distance between the bodies are constant. | background 1.00 | 0.06 | 0.02 | asserted 1.00 |
| s1 | In other words, the bodies undergo a rigid motion. | background 0.80 | 0.07 | 0.02 | asserted 1.00 |
| s2 | Here we investigate the possibility of partially rigid motions, where some but not all of the distances are constant. | contribution 0.89 | 0.48 | 0.03 | hedged possibility 0.67 |
| s3 | In particular, a {\em hinged} solution is one such that exactly one mutual distance varies. | background 0.59 | 0.26 | 0.04 | asserted 0.99 |
| s4 | The goal of this paper is to show that hinged solutions don't exist when n=3 or n=4. | contribution 0.60 | 0.73 | 0.93 | asserted 1.00 |
| s5 | For n=3 this means that if 2 of the 3 distances are constant so is the third and for n=4, if 5 of the 6 distances are constant, so is the sixth. | interpretation 0.58 | 0.81 | 0.33 | asserted 1.00 |
| s6 | These results hold independent of the dimension d of the ambient space. | result 0.70 | 0.81 | 0.05 | asserted 1.00 |
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Original abstract
A solution of the n-body problem in R^d is a relative equilibrium if all of the mutual distance between the bodies are constant. In other words, the bodies undergo a rigid motion. Here we investigate the possibility of partially rigid motions, where some but not all of the distances are constant. In particular, a {\em hinged} solution is one such that exactly one mutual distance varies. The goal of this paper is to show that hinged solutions don't exist when n=3 or n=4. For n=3 this means that if 2 of the 3 distances are constant so is the third and for n=4, if 5 of the 6 distances are constant, so is the sixth. These results hold independent of the dimension d of the ambient space.
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