{"id":"https://openalex.org/W2994428865","doi":"https://doi.org/10.1088/1361-6544/ab4d5f","title":"On the uniqueness of the isosceles trapezoidal central configuration in the 4-body problem for power-law potentials","display_name":"On the uniqueness of the isosceles trapezoidal central configuration in the 4-body problem for power-law potentials","publication_year":2019,"publication_date":"2019-12-02","ids":{"openalex":"https://openalex.org/W2994428865","doi":"https://doi.org/10.1088/1361-6544/ab4d5f","mag":"2994428865"},"language":"en","primary_location":{"id":"doi:10.1088/1361-6544/ab4d5f","doi":"https://doi.org/10.1088/1361-6544/ab4d5f","is_oa":true,"landing_page_url":"https://doi.org/10.1088/1361-6544/ab4d5f","pdf_url":"https://iopscience.iop.org/article/10.1088/1361-6544/ab4d5f/pdf","source":{"id":"https://openalex.org/S158842918","display_name":"Nonlinearity","issn_l":"0951-7715","issn":["0951-7715","1361-6544"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-b","jufo-1","norway-2","poland-100","russia-white-list-2"],"host_organization":"https://openalex.org/P4310320083","host_organization_name":"IOP Publishing","host_organization_lineage":["https://openalex.org/P4310320083","https://openalex.org/P4310311669"],"host_organization_lineage_names":["IOP Publishing","Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Nonlinearity","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://iopscience.iop.org/article/10.1088/1361-6544/ab4d5f/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057443007","display_name":"Ant\u00f4nio Carlos Fernandes","orcid":"https://orcid.org/0000-0001-9297-5101","observed_orcids":["https://orcid.org/0000-0001-9297-5101"]},"institutions":[{"id":"https://openalex.org/I8165072","display_name":"Universidade Federal de Itajub\u00e1","ror":"https://ror.org/00235nr42","country_code":"BR","type":"education","lineage":["https://openalex.org/I8165072"]}],"countries":["BR"],"is_corresponding":true,"raw_author_name":"Antonio Carlos Fernandes","raw_affiliation_strings":["Instituto de Matem\u00e1tica e Computa\u00e7\u00e3o, Universidade Federal de Itajub\u00e1, Avenida BPS, 1303, Pinheirinho, CEP: 37.500-903, Itajub\u00e1, MG, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Instituto de Matem\u00e1tica e Computa\u00e7\u00e3o, Universidade Federal de Itajub\u00e1, Avenida BPS, 1303, Pinheirinho, CEP: 37.500-903, Itajub\u00e1, MG, Brazil","institution_ids":["https://openalex.org/I8165072"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5003957478","display_name":"Luis Fernando Mello","orcid":"https://orcid.org/0000-0002-4989-3052","observed_orcids":["https://orcid.org/0000-0002-4989-3052"]},"institutions":[{"id":"https://openalex.org/I8165072","display_name":"Universidade Federal de Itajub\u00e1","ror":"https://ror.org/00235nr42","country_code":"BR","type":"education","lineage":["https://openalex.org/I8165072"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Luis Fernando Mello","raw_affiliation_strings":["Instituto de Matem\u00e1tica e Computa\u00e7\u00e3o, Universidade Federal de Itajub\u00e1, Avenida BPS, 1303, Pinheirinho, CEP: 37.500-903, Itajub\u00e1, MG, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Instituto de Matem\u00e1tica e Computa\u00e7\u00e3o, Universidade Federal de Itajub\u00e1, Avenida BPS, 1303, Pinheirinho, CEP: 37.500-903, Itajub\u00e1, MG, Brazil","institution_ids":["https://openalex.org/I8165072"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5026593361","display_name":"Cl\u00e1udio Vidal","orcid":"https://orcid.org/0000-0002-1630-0898","observed_orcids":["https://orcid.org/0000-0002-1630-0898"]},"institutions":[{"id":"https://openalex.org/I221450849","display_name":"University of B\u00edo-B\u00edo","ror":"https://ror.org/04dndfk38","country_code":"CL","type":"education","lineage":["https://openalex.org/I221450849"]}],"countries":["CL"],"is_corresponding":true,"raw_author_name":"Claudio Vidal","raw_affiliation_strings":["Grupo de Investigaci\u00f3n en Sistemas Din\u00e1micos y Aplicaciones-GISDA, Departamento de Matem\u00e1tica, Facultad de Ciencias, Universidad del B\u00edo-B\u00edo, Casilla 5-C, Concepci\u00f3n, VIII-Regi\u00f3n, Chile"],"raw_orcid":"https://orcid.org/0000-0002-1630-0898","affiliations":[{"raw_affiliation_string":"Grupo de Investigaci\u00f3n en Sistemas Din\u00e1micos y Aplicaciones-GISDA, Departamento de Matem\u00e1tica, Facultad de Ciencias, Universidad del B\u00edo-B\u00edo, Casilla 5-C, Concepci\u00f3n, VIII-Regi\u00f3n, Chile","institution_ids":["https://openalex.org/I221450849"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5026593361","https://openalex.org/A5057443007"],"corresponding_institution_ids":["https://openalex.org/I221450849","https://openalex.org/I8165072"],"apc_list":{"value":2700,"currency":"USD","value_usd":2700},"apc_paid":null,"fwci":1.0228,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.75082846,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"33","issue":"1","first_page":"388","last_page":"407"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9702000021934509,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9702000021934509,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11394","display_name":"Dynamics and Control of Mechanical Systems","score":0.010400000028312206,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11082","display_name":"Spacecraft Dynamics and Control","score":0.004100000020116568,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/central-configurations","display_name":"central configurations","score":0.925000011920929},{"id":"https://openalex.org/keywords/isosceles-trapezoid","display_name":"isosceles trapezoid","score":0.8970000147819519},{"id":"https://openalex.org/keywords/four-body-problem","display_name":"four-body problem","score":0.8610000014305115},{"id":"https://openalex.org/keywords/power-law-potential","display_name":"power-law potential","score":0.9649999737739563},{"id":"https://openalex.org/keywords/newtonian-gravity","display_name":"Newtonian gravity","score":0.7009999752044678},{"id":"https://openalex.org/keywords/uniqueness","display_name":"uniqueness","score":0.6869999766349792},{"id":"https://openalex.org/keywords/equal-masses","display_name":"equal masses","score":0.5870000123977661}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8586610555648804},{"id":"https://openalex.org/C98653994","wikidata":"https://www.wikidata.org/wiki/Q875937","display_name":"Isosceles triangle","level":2,"score":0.8381392955780029},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.6848894357681274},{"id":"https://openalex.org/C151756577","wikidata":"https://www.wikidata.org/wiki/Q36810","display_name":"Quadrilateral","level":3,"score":0.6783517003059387},{"id":"https://openalex.org/C70920666","wikidata":"https://www.wikidata.org/wiki/Q4328383","display_name":"Newtonian potential","level":3,"score":0.5255920886993408},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4963410496711731},{"id":"https://openalex.org/C87040749","wikidata":"https://www.wikidata.org/wiki/Q428971","display_name":"Power law","level":2,"score":0.42451170086860657},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.4217611253261566},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.41603580117225647},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.37515369057655334},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.31430578231811523},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.11100181937217712},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07038739323616028},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C124017977","wikidata":"https://www.wikidata.org/wiki/Q11412","display_name":"Gravitation","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1088/1361-6544/ab4d5f","doi":"https://doi.org/10.1088/1361-6544/ab4d5f","is_oa":true,"landing_page_url":"https://doi.org/10.1088/1361-6544/ab4d5f","pdf_url":"https://iopscience.iop.org/article/10.1088/1361-6544/ab4d5f/pdf","source":{"id":"https://openalex.org/S158842918","display_name":"Nonlinearity","issn_l":"0951-7715","issn":["0951-7715","1361-6544"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-b","jufo-1","norway-2","poland-100","russia-white-list-2"],"host_organization":"https://openalex.org/P4310320083","host_organization_name":"IOP Publishing","host_organization_lineage":["https://openalex.org/P4310320083","https://openalex.org/P4310311669"],"host_organization_lineage_names":["IOP Publishing","Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Nonlinearity","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1088/1361-6544/ab4d5f","doi":"https://doi.org/10.1088/1361-6544/ab4d5f","is_oa":true,"landing_page_url":"https://doi.org/10.1088/1361-6544/ab4d5f","pdf_url":"https://iopscience.iop.org/article/10.1088/1361-6544/ab4d5f/pdf","source":{"id":"https://openalex.org/S158842918","display_name":"Nonlinearity","issn_l":"0951-7715","issn":["0951-7715","1361-6544"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-b","jufo-1","norway-2","poland-100","russia-white-list-2"],"host_organization":"https://openalex.org/P4310320083","host_organization_name":"IOP Publishing","host_organization_lineage":["https://openalex.org/P4310320083","https://openalex.org/P4310311669"],"host_organization_lineage_names":["IOP Publishing","Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Nonlinearity","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[{"score":0.5199999809265137,"display_name":"Peace, justice, and strong institutions","id":"https://metadata.un.org/sdg/16"}],"study_designs":[],"awards":[{"id":"https://openalex.org/G247810146","display_name":null,"funder_award_id":"2018\u20104","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"},{"id":"https://openalex.org/G2890287661","display_name":"GEOMETRIA DE CURVAS PLANAS, SISTEMAS HAMILTONIANOS E SISTEMAS DESCONT\u00cdNUOS","funder_award_id":"APQ-01158-17","funder_id":"https://openalex.org/F4320322980","funder_display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de Minas Gerais"},{"id":"https://openalex.org/G5412455698","display_name":"ESTUDO DE CONFIGURA\u00c7\u00d5ES CENTRAIS E SOLU\u00c7\u00d5ES HOMOGR\u00c1FICAS CELESTE","funder_award_id":"APQ-03149-18","funder_id":"https://openalex.org/F4320322980","funder_display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de Minas Gerais"}],"funders":[{"id":"https://openalex.org/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https://ror.org/03swz6y49"},{"id":"https://openalex.org/F4320322980","display_name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de Minas Gerais","ror":"https://ror.org/00nc55f03"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2994428865.pdf","grobid_xml":"https://content.openalex.org/works/W2994428865.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W579379960","https://openalex.org/W1506377182","https://openalex.org/W1555780618","https://openalex.org/W1968524335","https://openalex.org/W1975422833","https://openalex.org/W1999126009","https://openalex.org/W2007301691","https://openalex.org/W2020575516","https://openalex.org/W2026434926","https://openalex.org/W2044413922","https://openalex.org/W2050338848","https://openalex.org/W2050914063","https://openalex.org/W2051109533","https://openalex.org/W2053967969","https://openalex.org/W2059552911","https://openalex.org/W2062713134","https://openalex.org/W2068310052","https://openalex.org/W2072764815","https://openalex.org/W2080618918","https://openalex.org/W2088172592","https://openalex.org/W2116493231","https://openalex.org/W2122320406","https://openalex.org/W2122696655","https://openalex.org/W2156971036","https://openalex.org/W2163217780","https://openalex.org/W2172066699","https://openalex.org/W2270342017","https://openalex.org/W2335595795","https://openalex.org/W2473060496","https://openalex.org/W2518000031","https://openalex.org/W2518037270","https://openalex.org/W2529659539","https://openalex.org/W2622044824","https://openalex.org/W3105021676","https://openalex.org/W3105025644","https://openalex.org/W3105945846","https://openalex.org/W3179340936","https://openalex.org/W3215067209","https://openalex.org/W4205891059"],"related_works":["https://openalex.org/W2795405095","https://openalex.org/W2263342410","https://openalex.org/W2991387667","https://openalex.org/W4244438377","https://openalex.org/W2795954609","https://openalex.org/W134513340","https://openalex.org/W4287901079","https://openalex.org/W2299987007","https://openalex.org/W2795693449","https://openalex.org/W4388482586"],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2],"article":[3],"we":[4,68,97],"study":[5,45,95],"the":[6,15,18,36,71,75,83],"planar":[7],"4-body":[8],"problem":[9,38],"under":[10],"homogeneous":[11],"power-law":[12],"potentials":[13],"where":[14],"interaction":[16],"between":[17],"bodies":[19],"is":[20,74,91],"given":[21],"by":[22],"r":[23],"\u2212":[24],"a":[25,32,41,62,87],",":[26],"(the":[27],"Newtonian":[28],"case":[29,84],"corresponding":[30,39],"to":[31,40,94],"=":[33,42],"3":[34],"and":[35],"vortex":[37],"2).":[43],"We":[44],"convex":[46,63],"central":[47,77],"configurations":[48],"assuming":[49],"two":[50,58],"pairs":[51],"of":[52,61],"positive":[53],"equal":[54],"masses":[55],"located":[56],"at":[57],"adjacent":[59],"vertices":[60],"quadrilateral.":[64],"Under":[65],"these":[66],"assumptions":[67],"prove":[69],"that":[70],"isosceles":[72],"trapezoid":[73],"unique":[76],"configuration":[78],"for":[79],"every":[80],".":[81],"For":[82],"0":[85],"<":[86,88],"4/3,":[89],"which":[90],"more":[92],"difficult":[93],"analytically,":[96],"present":[98],"some":[99],"numerical":[100],"considerations.":[101]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":3}],"updated_date":"2026-10-14T16:39:01.820549","created_date":"2025-10-10T00:00:00"}