Partially rigid motions in the n-body problem

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...

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Bibliographic Details
Published in:arXiv.org 2024-07
Main Author: Moeckel, Richard
Format: Article
Language:English
Subjects:
Many body problem
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Summary: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.
ISSN:2331-8422