{"id":"https://openalex.org/W7214932307","doi":"https://doi.org/10.48550/arxiv.2609.36990","title":"Post-Newtonian secular dynamics of hierarchical triples. I. Eccentricity-, inclination- and node-dependence of the multiple-scale formulation, and its canonical consistency","display_name":"Post-Newtonian secular dynamics of hierarchical triples. I. Eccentricity-, inclination- and node-dependence of the multiple-scale formulation, and its canonical consistency","publication_year":2026,"publication_date":"2026-09-29","ids":{"openalex":"https://openalex.org/W7214932307","doi":"https://doi.org/10.48550/arxiv.2609.36990"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.36990","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.36990","pdf_url":"https://arxiv.org/pdf/2609.36990","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.36990","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058627673","display_name":"Hideyoshi Arakida","orcid":"https://orcid.org/0000-0002-0319-9305","observed_orcids":["https://orcid.org/0000-0002-0319-9305"]},"institutions":[{"id":"https://openalex.org/I104946051","display_name":"Nihon University","ror":"https://ror.org/05jk51a88","country_code":"JP","type":"education","lineage":["https://openalex.org/I104946051"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Arakida, Hideyoshi","raw_affiliation_strings":["College of Engineering , Nihon University , 1 Nakagawara , Tokusada , Tamuramachi , Koriyama , Fukushima 963-8642 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Engineering , Nihon University , 1 Nakagawara , Tokusada , Tamuramachi , Koriyama , Fukushima 963-8642 , Japan","institution_ids":["https://openalex.org/I104946051"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5058627673"],"corresponding_institution_ids":["https://openalex.org/I104946051"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10039","display_name":"Stellar, planetary, and galactic studies","score":0.5654000043869019,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10039","display_name":"Stellar, planetary, and galactic studies","score":0.5654000043869019,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10463","display_name":"Pulsars and Gravitational Waves Research","score":0.26170000433921814,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11960","display_name":"Relativity and Gravitational Theory","score":0.05790000036358833,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hierarchical-triples","display_name":"hierarchical triples","score":0.8090000152587891},{"id":"https://openalex.org/keywords/post-newtonian-dynamics","display_name":"post-Newtonian dynamics","score":0.9100000262260437},{"id":"https://openalex.org/keywords/secular-equation","display_name":"secular equation","score":0.5699999928474426},{"id":"https://openalex.org/keywords/multiscale-expansion","display_name":"multiscale expansion","score":0.9440000057220459},{"id":"https://openalex.org/keywords/hamiltonicity","display_name":"Hamiltonicity","score":0.6779999732971191},{"id":"https://openalex.org/keywords/effective-field-theory","display_name":"effective field theory","score":0.6470000147819519},{"id":"https://openalex.org/keywords/true-anomaly","display_name":"true anomaly","score":0.6200000047683716},{"id":"https://openalex.org/keywords/eccentric-anomaly","display_name":"eccentric anomaly","score":0.7770000100135803},{"id":"https://openalex.org/keywords/octupole","display_name":"octupole","score":0.6850000023841858},{"id":"https://openalex.org/keywords/orbital-evolution","display_name":"orbital evolution","score":0.39800000190734863}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5879999995231628},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5566999912261963},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.541100025177002},{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.49709999561309814},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.45329999923706055},{"id":"https://openalex.org/C39984356","wikidata":"https://www.wikidata.org/wiki/Q2121987","display_name":"Quadrupole","level":2,"score":0.42640000581741333},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.38839998841285706},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.3720000088214874},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3621000051498413},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.35120001435279846},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33629998564720154},{"id":"https://openalex.org/C5555538","wikidata":"https://www.wikidata.org/wiki/Q6823703","display_name":"Method of averaging","level":3,"score":0.3237000107765198},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3208000063896179},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.30309998989105225},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.2973000109195709},{"id":"https://openalex.org/C12997251","wikidata":"https://www.wikidata.org/wiki/Q567560","display_name":"Anomaly (physics)","level":2,"score":0.29409998655319214},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.29170000553131104},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.2800000011920929},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.2711000144481659},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.2676999866962433}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.36990","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.36990","pdf_url":"https://arxiv.org/pdf/2609.36990","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.36990","doi":"https://doi.org/10.48550/arxiv.2609.36990","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.36990","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.36990","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.36990","pdf_url":"https://arxiv.org/pdf/2609.36990","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320309995","display_name":"Nihon University","ror":"https://ror.org/05jk51a88"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214932307.pdf","grobid_xml":"https://content.openalex.org/works/W7214932307.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W2093149011","https://openalex.org/W2102640447","https://openalex.org/W2307344727","https://openalex.org/W2309338676","https://openalex.org/W3000281460","https://openalex.org/W3177884199","https://openalex.org/W4319313220","https://openalex.org/W4362705079","https://openalex.org/W4406457564","https://openalex.org/W4410638896"],"related_works":["https://openalex.org/W7221834547","https://openalex.org/W2045666801","https://openalex.org/W2513271499","https://openalex.org/W6891997890","https://openalex.org/W4406457564","https://openalex.org/W2980705355","https://openalex.org/W1974872970","https://openalex.org/W2104213524","https://openalex.org/W4402028357","https://openalex.org/W7096728338"],"abstract_inverted_index":{"We":[0],"will":[1],"re-examine":[2],"the":[3,24,29,36,46,52,58,66,69,79,86,97,102,122,138,154,176,190,207,218,226,235],"leading":[4],"post-Newtonian":[5],"(1PN)":[6],"cross":[7],"terms":[8,130],"of":[9,39,106,121,124,175,187,228],"hierarchical":[10],"triples":[11],"obtained":[12],"by":[13,223],"Lim":[14],"and":[15,28,42,49,140,198,205],"Rodriguez":[16],"(LR)":[17],"with":[18,35,76,101,115,148,167,203,212,234],"a":[19,107,125,149,163],"two-parameter":[20],"multiple-scale":[21],"expansion":[22],"in":[23,65,93,113],"hierarchy":[25],"parameter":[26,31],"$\\varepsilon=a_1/a_2$":[27],"1PN":[30,170],"$\u03b4$,":[32],"which":[33,133],"disagree":[34],"effective-field-theory":[37],"result":[38],"Kuntz,":[40],"Serra,":[41],"Trincherini":[43],"(KST)":[44],"at":[45],"orders":[47],"$\u03b4\\varepsilon^{3/2}$":[48,59],"$\u03b4\\varepsilon^{7/2}$.":[50],"Reconstructing":[51],"LR":[53,188,224],"calculation,":[54],"we":[55],"show":[56,216],"that":[57,217],"``libration''":[60],"term":[61,120,186],"[LR":[62],"Eq.~(4.5)]":[63],"originates":[64],"averaging":[67],"measure:":[68],"quadrupole":[70,165],"periodic":[71],"solutions":[72],"are":[73,131],"made":[74],"mean-free":[75],"respect":[77],"to":[78,181],"outer":[80],"true":[81],"anomaly":[82],"but":[83],"re-substituted":[84],"under":[85],"time":[87,103],"average.":[88],"The":[89,184],"resulting":[90],"term,":[91],"evaluated":[92],"closed":[94],"form,":[95],"reproduces":[96],"implemented":[98],"one":[99,119],"exactly;":[100],"measure":[104],"required":[105],"canonical":[108],"generating":[109],"function":[110],"it":[111,136],"vanishes,":[112],"agreement":[114],"KST.":[116],"Equation~(4.5)":[117],"is":[118,134],"flow":[123],"removable":[126],"Hamiltonian":[127],"whose":[128],"companion":[129],"absent,":[132],"why":[135],"fails":[137],"rationality":[139],"Hamiltonicity":[141,208],"tests.":[142],"A":[143],"calculation":[144],"carried":[145],"out":[146],"consistently":[147],"single":[150],"phase":[151],"variable":[152],"yields":[153],"complete,":[155],"gauge-equivalent":[156],"secular":[157,214],"equations,":[158],"whereas":[159,230],"LR's":[160,213],"implementation":[161,177],"combines":[162],"true-anomaly":[164],"solution":[166],"an":[168],"eccentric-anomaly":[169],"solution.":[171],"Two":[172],"structural":[173],"defects":[174],"do":[178],"not":[179],"contribute":[180],"these":[182],"terms.":[183],"$\u03b4\\varepsilon^{7/2}$":[185],"has":[189],"correct":[191],"mass":[192],"dependence":[193],"only":[194],"for":[195],"$m\\ll":[196],"m_3$,":[197],"its":[199],"angular":[200],"structure":[201],"disagrees":[202],"KST":[204],"violates":[206],"test.":[209],"Finally,":[210],"integrations":[211],"code":[215],"resonant":[219],"ZLK":[220],"modulations":[221],"reported":[222],"survive":[225],"removal":[227],"Eq.~(4.5),":[229],"their":[231],"accelerated":[232],"merger":[233],"octupole":[236],"disappears.":[237]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-01T00:00:00"}
