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It is shown that the Morse index changes at a bifurcation point and all solutions bifurcating are approximated by variational functions responsible for the change of the Morse index. Inversely we observed %numerically bifurcation occurs at every point where the Morse index changes for the figure-eight choreo"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.00115","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2019-01-01T08:28:59Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"84ff5714b44a7e2937f079dfebdbc1016583feb490d933a3b27db328bbfd5a1d","abstract_canon_sha256":"0f3a8b639745e59bbbca8c5d1e90c7b3947d43932703aeb02b6d30ab22332c8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:57.964747Z","signature_b64":"sQt8oUfR+Ln2EzLxJMMM0ravL3rNPBx4Uy7TZCC2Beu/zKH8U6bO3G6T+fK/2Rx0LmicaNRlQE8fss/uXW07Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed5fb1ef79dc90aadf8344db681bbf74463476c0a085b41e4c5a4f07c171942e","last_reissued_at":"2026-05-17T23:48:57.964221Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:57.964221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Hiroshi Fukuda, Hiroshi Ozaki, Toshiaki Fujiwara","submitted_at":"2019-01-01T08:28:59Z","abstract_excerpt":"We report on the Morse index and periodic solutions bifurcating from the figure-eight choreography for the equal mass three-body problem under homogeneous potential $-1/r^a$ for $a \\ge 0$, and under Lennard-Jones (LJ) type potential $1/r^{12}-1/r^6$, where $r$ is a distance between bodies. 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