Pair Space in Classical Mechanics I. The Three-Body Problem
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910657107984384 |
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| author | Drory, Alon |
| author_facet | Drory, Alon |
| contents | I introduce an extended configuration space for classical mechanical systems, called pair-space, which is spanned by the relative positions of all the pairs of bodies. To overcome the non-independence of this basis, one adds to the Lagrangian a term containing auxiliary variables. As a proof of concept, I apply this representation to the three-body problem with a generalized potential that depends on the distance $r$ between the bodies as $r^{-n}$. I obtain the equilateral and collinear solutions (corresponding to the Lagrange and Euler solutions if $n=1$) in a particularly simple way. In the collinear solution, this representation leads to several new bounds on the relative distances of the bodies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14737 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pair Space in Classical Mechanics I. The Three-Body Problem Drory, Alon Mathematical Physics 70F10, 70F15, 70H03, 70F07 I introduce an extended configuration space for classical mechanical systems, called pair-space, which is spanned by the relative positions of all the pairs of bodies. To overcome the non-independence of this basis, one adds to the Lagrangian a term containing auxiliary variables. As a proof of concept, I apply this representation to the three-body problem with a generalized potential that depends on the distance $r$ between the bodies as $r^{-n}$. I obtain the equilateral and collinear solutions (corresponding to the Lagrange and Euler solutions if $n=1$) in a particularly simple way. In the collinear solution, this representation leads to several new bounds on the relative distances of the bodies. |
| title | Pair Space in Classical Mechanics I. The Three-Body Problem |
| topic | Mathematical Physics 70F10, 70F15, 70H03, 70F07 |
| url | https://arxiv.org/abs/2410.14737 |