Pair Space in Classical Mechanics II. N-Body Central Configurations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910657101692928 |
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| author | Drory, Alon |
| author_facet | Drory, Alon |
| contents | A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the $N$-body Newtonian problem, a configuration that does not remain on a fixed line in space is a central configuration if and only if it conserves all pair angular momenta. For collinear systems, I obtain a set of equations for the ratios of the relative distances of the bodies, from which I derive some bounds on the minimal length of the line. For the non-collinear case I derive some geometrical relations, independent of the masses of the bodies. These are necessary conditions for a non-collinear configuration to be central. They generalize, to arbitrary $N$, a consequence of the Dziobek relation, which holds for $N=4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14736 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pair Space in Classical Mechanics II. N-Body Central Configurations Drory, Alon Mathematical Physics Classical Physics 70F10, 70F15, 70H03 A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the $N$-body Newtonian problem, a configuration that does not remain on a fixed line in space is a central configuration if and only if it conserves all pair angular momenta. For collinear systems, I obtain a set of equations for the ratios of the relative distances of the bodies, from which I derive some bounds on the minimal length of the line. For the non-collinear case I derive some geometrical relations, independent of the masses of the bodies. These are necessary conditions for a non-collinear configuration to be central. They generalize, to arbitrary $N$, a consequence of the Dziobek relation, which holds for $N=4$. |
| title | Pair Space in Classical Mechanics II. N-Body Central Configurations |
| topic | Mathematical Physics Classical Physics 70F10, 70F15, 70H03 |
| url | https://arxiv.org/abs/2410.14736 |