{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T21:30:48Z","timestamp":1783114248633,"version":"3.54.6"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T00:00:00Z","timestamp":1727395200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T00:00:00Z","timestamp":1727395200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s00332-024-10087-7","type":"journal-article","created":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T15:02:44Z","timestamp":1727449364000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Schubart Orbits in the Curved Three-Body Problem with Two Equal Masses"],"prefix":"10.1007","volume":"34","author":[{"given":"Shuqiang","family":"Zhu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,9,27]]},"reference":[{"key":"10087_CR1","first-page":"37","volume":"89","author":"A Albouy","year":"2013","unstructured":"Albouy, A.: There is a projective dynamics. Eur. Math. Soc. Newsl. 89, 37\u201343 (2013)","journal-title":"Eur. Math. Soc. Newsl."},{"issue":"6","key":"10087_CR2","first-page":"1","volume":"130","author":"AV Borisov","year":"2018","unstructured":"Borisov, A.V., Garc\u00eda-Naranjo, L.C., Mamaev, I.S., Montaldi, J.: Reduction and relative equilibria for the two-body problem on spaces of constant curvature. Celest. Mech. Dyn. Astronom. 130(6), 1\u201336 (2018)","journal-title":"Celest. Mech. Dyn. Astronom."},{"issue":"3","key":"10087_CR3","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1070\/RD2004v009n03ABEH000280","volume":"9","author":"AV Borisov","year":"2004","unstructured":"Borisov, A.V., Mamaev, I.S., Kilin, A.A.: Two-body problem on a sphere. Reduction, stochasticity, periodic orbits. Regul. Chaotic Dyn. 9(3), 265\u2013279 (2004)","journal-title":"Regul. Chaotic Dyn."},{"key":"10087_CR4","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1090\/S0002-9947-1971-0279830-1","volume":"158","author":"C Conley","year":"1971","unstructured":"Conley, C., Easton, R.: Isolated invariant sets and isolating blocks. Trans. Am. Math. Soc. 158, 35\u201361 (1971)","journal-title":"Trans. Am. Math. Soc."},{"key":"10087_CR5","series-title":"Isolated invariant sets and the Morse index","volume-title":"CBMS Regional Conference Series in Mathematics","author":"C Conley","year":"1978","unstructured":"Conley, C.: CBMS Regional Conference Series in Mathematics. Isolated invariant sets and the Morse index, vol. 38. American Mathematical Society, Providence, R.I. (1978)"},{"key":"10087_CR6","doi-asserted-by":"publisher","first-page":"5851","DOI":"10.1016\/j.jde.2019.06.008","volume":"267","author":"Y Deng","year":"2019","unstructured":"Deng, Y., Diacu, F., Zhu, S.: Variational property of periodic Kepler orbits in constant curvature spaces. J. Differ. Equ. 267, 5851\u20135869 (2019)","journal-title":"J. Differ. Equ."},{"issue":"4","key":"10087_CR7","doi-asserted-by":"publisher","first-page":"2249","DOI":"10.1090\/S0002-9947-2010-05251-1","volume":"363","author":"F Diacu","year":"2011","unstructured":"Diacu, F.: On the singularities of the curved n-body problem. Trans. Am. Math. Soc. 363(4), 2249\u20132264 (2011)","journal-title":"Trans. Am. Math. Soc."},{"issue":"1071","key":"10087_CR8","first-page":"vi+80","volume":"228","author":"F Diacu","year":"2014","unstructured":"Diacu, F.: Relative equilibria in the 3-dimensional curved $$n$$-body problem. Mem. Am. Math. Soc. 228(1071), vi+80 (2014)","journal-title":"Mem. Am. Math. Soc."},{"issue":"2","key":"10087_CR9","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/s00332-011-9117-y","volume":"22","author":"F Diacu","year":"2012","unstructured":"Diacu, F., P\u00e9rez-Chavela, E., Santoprete, M.: The n-body problem in spaces of constant curvature. part ii: singularities. J. Nonlinear Sci. 22(2), 267\u2013275 (2012)","journal-title":"J. Nonlinear Sci."},{"issue":"5","key":"10087_CR10","doi-asserted-by":"publisher","first-page":"1999","DOI":"10.1007\/s00332-018-9473-y","volume":"28","author":"F Diacu","year":"2018","unstructured":"Diacu, F., Stoica, C., Zhu, S.: Central configurations of the curved n-body problem. J. Nonlinear Sci. 28(5), 1999\u20132046 (2018)","journal-title":"J. Nonlinear Sci."},{"issue":"31","key":"10087_CR11","doi-asserted-by":"publisher","first-page":"315101","DOI":"10.1088\/1751-8121\/aaca41","volume":"51","author":"V Dmitra\u0161inovi\u0107","year":"2018","unstructured":"Dmitra\u0161inovi\u0107, V., Hudomal, A., Shibayama, M., Sugita, A.: Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler\u2019s third law: a numerical test. J. Phys. A 51(31), 315101 (2018)","journal-title":"J. Phys. A"},{"key":"10087_CR12","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1016\/0022-0396(70)90123-3","volume":"7","author":"RW Easton","year":"1970","unstructured":"Easton, R.W.: On the existence of invariant sets inside a submanifold convex to a flow. J. Differ. Equ. 7, 54\u201368 (1970)","journal-title":"J. Differ. Equ."},{"issue":"4\u20135","key":"10087_CR13","doi-asserted-by":"publisher","first-page":"690","DOI":"10.1134\/S1560354723040111","volume":"28","author":"T Fujiwara","year":"2023","unstructured":"Fujiwara, T., P\u00e9rez-Chavela, E.: Three-body relative equilibria on $${\\mathbb{S} }^2$$. Regul. Chaotic Dyn. 28(4\u20135), 690\u2013706 (2023)","journal-title":"Regul. Chaotic Dyn."},{"issue":"4","key":"10087_CR14","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1007\/BF00049149","volume":"54","author":"VV Kozlov","year":"1992","unstructured":"Kozlov, V.V., Harin, A.O.: Kepler\u2019s problem in constant curvature spaces. Celest. Mech. Dyn. Astronom. 54(4), 393\u2013399 (1992)","journal-title":"Celest. Mech. Dyn. Astronom."},{"issue":"12","key":"10087_CR15","doi-asserted-by":"publisher","first-page":"4639","DOI":"10.1088\/1361-6544\/ab360d","volume":"32","author":"W Kuang","year":"2019","unstructured":"Kuang, W., Ouyang, T., Xie, Z., Yan, D.: The broucke-h\u00e9non orbit and the schubart orbit in the planar three-body problem with two equal masses. Nonlinearity 32(12), 4639 (2019)","journal-title":"Nonlinearity"},{"key":"10087_CR16","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1007\/BF01390175","volume":"27","author":"R McGehee","year":"1974","unstructured":"McGehee, R.: Triple collision in the collinear three-body problem. Invent. Math. 27, 191\u2013227 (1974)","journal-title":"Invent. Math."},{"issue":"2\u20133","key":"10087_CR17","doi-asserted-by":"publisher","first-page":"609","DOI":"10.3934\/dcdsb.2008.10.609","volume":"10","author":"R Moeckel","year":"2008","unstructured":"Moeckel, R.: A topological existence proof for the Schubart orbits in the collinear three-body problem. Discret. Contin. Dyn. Syst. Ser. B 10(2\u20133), 609\u2013620 (2008)","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"issue":"1","key":"10087_CR18","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1137\/15M1024652","volume":"15","author":"H Montanelli","year":"2016","unstructured":"Montanelli, H., Gushterov, N.I.: Computing planar and spherical choreographies. SIAM J. Appl. Dyn. Syst. 15(1), 235\u2013256 (2016)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"10087_CR19","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1002\/asna.19562830105","volume":"283","author":"J Schubart","year":"1956","unstructured":"Schubart, J.: Numerische Aufsuchung periodischer L\u00f6sungen im Dreik\u00f6rperproblem. Astronom. Nachr. 283, 17\u201322 (1956)","journal-title":"Astronom. Nachr."},{"issue":"3","key":"10087_CR20","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1007\/s00205-010-0334-6","volume":"199","author":"M Shibayama","year":"2011","unstructured":"Shibayama, M.: Minimizing periodic orbits with regularizable collisions in the $$n$$-body problem. Arch. Ration. Mech. Anal. 199(3), 821\u2013841 (2011)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2\u20133","key":"10087_CR21","doi-asserted-by":"publisher","first-page":"699","DOI":"10.3934\/dcdsb.2008.10.699","volume":"10","author":"A Venturelli","year":"2008","unstructured":"Venturelli, A.: A variational proof of the existence of von Schubart\u2019s orbit. Discret. Contin. Dyn. Syst. Ser. B 10(2\u20133), 699\u2013717 (2008)","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"issue":"279\u2013313","key":"10087_CR22","first-page":"1947","volume":"20","author":"T Wa\u017cewski","year":"1948","unstructured":"Wa\u017cewski, T.: Sur un principe topologique de l\u2019examen de l\u2019allure asymptotique des int\u00e9grales des \u00e9quations diff\u00e9rentielles ordinaires. Ann. Soc. Polon. Math. 20(279\u2013313), 1947 (1948)","journal-title":"Ann. Soc. Polon. Math."},{"key":"10087_CR23","series-title":"Introduction to applied nonlinear dynamical systems and chaos","volume-title":"Texts in Applied Mathematics","author":"S Wiggins","year":"1990","unstructured":"Wiggins, S.: Texts in Applied Mathematics. Introduction to applied nonlinear dynamical systems and chaos, vol. 2. Springer-Verlag, New York (1990)"},{"issue":"1","key":"10087_CR24","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/s11464-012-0171-z","volume":"7","author":"D Yan","year":"2012","unstructured":"Yan, D.: A simple existence proof of Schubart periodic orbit with arbitrary masses. Front. Math. China 7(1), 145\u2013160 (2012)","journal-title":"Front. Math. China"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-024-10087-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-024-10087-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-024-10087-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T03:08:30Z","timestamp":1730171310000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-024-10087-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,27]]},"references-count":24,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["10087"],"URL":"https:\/\/doi.org\/10.1007\/s00332-024-10087-7","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,27]]},"assertion":[{"value":"9 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 August 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 September 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"109"}}