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Marchal-type lemmas for the restricted (N+1)-body problem with their applications

  • Kuo-chang Chen
  • Guowei Yu
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Abstract

The direct methods of calculus of variations have had a great success in the classical Newtonian N-body problem in this century. First in finding periodic solutions and more recently in the existence of parabolic, hyperbolic and hyperbolic-parabolic solutions. The foundation of many of these results is Marchal's lemma, which guarantee's fixed-end local minimizers are collision-free and hence classical solutions of the N-body problem. Meanwhile almost no result is available in the restricted problem through the direct methods of calculus of variations, mainly due to the lack of a Marchal-type lemma. In this paper such a lemma and its variation will be obtained for the restricted problem. We then apply them to the restricted problem to study unbounded (hyperbolic, parabolic, bi-hyperbolic and bi-parabolic) solutions, and bounded (periodic and quasi-periodic) solutions. We emphasize that the extension to the restricted case does not follow from the standard arguement for the unrestricted case.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026marchaltype,
  title = {{Marchal-type lemmas for the restricted $(N+1)$-body problem with their applications}},
  author = {Kuo-chang Chen and Guowei Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22849},
  eprint = {2610.22849},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22849},
}
RIS
TY  - UNPB
TI  - Marchal-type lemmas for the restricted (N+1)-body problem with their applications
AU  - Kuo-chang Chen
AU  - Guowei Yu
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.22849
UR  - https://arxiv.org/abs/2610.22849
AB  - The direct methods of calculus of variations have had a great success in the classical Newtonian N-body problem in this century. First in finding periodic solutions and more recently in the existence of parabolic, hyperbolic and hyperbolic-parabolic solutions. The foundation of many of these results is Marchal's lemma, which guarantee's fixed-end local minimizers are collision-free and hence classical solutions of the N-body problem. Meanwhile almost no result is available in the restricted problem through the direct methods of calculus of variations, mainly due to the lack of a Marchal-type lemma. In this paper such a lemma and its variation will be obtained for the restricted problem. We then apply them to the restricted problem to study unbounded (hyperbolic, parabolic, bi-hyperbolic and bi-parabolic) solutions, and bounded (periodic and quasi-periodic) solutions. We emphasize that the extension to the restricted case does not follow from the standard arguement for the unrestricted case.
ER  -