{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22849",
  "author_total": 2,
  "authors": [
    {
      "name": "Kuo-chang Chen"
    },
    {
      "name": "Guowei Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d0tzv4p6vk5rn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22849",
  "fulltext_url": null,
  "id": "d0tzv4p6vk5rn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22849",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2202|Spacecraft Dynamics and Control"
    ]
  ],
  "openalex": "W7222986914",
  "pdf_url": "https://arxiv.org/pdf/2610.22849",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d0tzv4p6vk5rn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.22849",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22849"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Marchal-type lemmas for the restricted $(N+1)$-body problem with their applications",
  "title_text": "Marchal-type lemmas for the restricted (N+1)-body problem with their applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/d0tzv4p6vk5rn",
  "venue": "arXiv",
  "versions": []
}