{"paper":{"title":"A continuum of periodic solutions to the four-body problem with various choices of masses","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Tiancheng Ouyang, Zhifu Xie","submitted_at":"2013-10-15T21:13:11Z","abstract_excerpt":"In this paper, we apply the variational method with the Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for planar four-body problem with $m_1=m_3$ and $m_2=m_4$. A path $q(t)$ in $[0,T]$ satisfies SPBC if the boundaries $q(0)\\in \\mathbf{A}$ and $q(T)\\in \\mathbf{B}$, where $\\mathbf{A}$ and $\\mathbf{B}$ are two structural configuration spaces in $(\\mathbf{R}^2)^4$ and they depend on a rotation angle $\\theta\\in (0,2\\pi)$ and the mass ratio $\\mu=\\frac{m_2}{m_1}\\in \\mathbf{R}^+$.\n  We show that there is a region $\\Omega\\subseteq (0,2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4206","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}