{"id":"https://openalex.org/W7204828480","doi":"https://doi.org/10.5281/zenodo.22209513","title":"Four Formulations and One Motion: A Narrative Review of Classical Mechanics' Methods and Instruments --- Newtonian, Lagrangian, Hamiltonian, and Hamilton-Jacobi","display_name":"Four Formulations and One Motion: A Narrative Review of Classical Mechanics' Methods and Instruments --- Newtonian, Lagrangian, Hamiltonian, and Hamilton-Jacobi","publication_year":2026,"publication_date":"2026-08-31","ids":{"openalex":"https://openalex.org/W7204828480","doi":"https://doi.org/10.5281/zenodo.22209513"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.22209513","doi":"https://doi.org/10.5281/zenodo.22209513","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22209513","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"JournalArticle"},"volume":null,"type":"review","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.22209513","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120933120","display_name":"Zen Revista","orcid":"https://orcid.org/0009-0007-6299-2008","observed_orcids":["https://orcid.org/0009-0007-6299-2008"]},"institutions":[{"id":"https://openalex.org/I4210124990","display_name":"Faculdades Guarulhos","ror":"https://ror.org/02s6maw65","country_code":"BR","type":"other","lineage":["https://openalex.org/I4210124990"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Revista, Zen","raw_affiliation_strings":["Faculdade de Guarulhos (FG)"],"raw_orcid":"https://orcid.org/0009-0007-6299-2008","affiliations":[{"raw_affiliation_string":"Faculdade de Guarulhos (FG)","institution_ids":["https://openalex.org/I4210124990"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5148776992","display_name":"10 PHYSICS","orcid":"https://orcid.org/0009-0007-6299-2008","observed_orcids":["https://orcid.org/0009-0007-6299-2008"]},"institutions":[{"id":"https://openalex.org/I4210124990","display_name":"Faculdades Guarulhos","ror":"https://ror.org/02s6maw65","country_code":"BR","type":"other","lineage":["https://openalex.org/I4210124990"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"PHYSICS, 10","raw_affiliation_strings":["Faculdade de Guarulhos (FG)"],"raw_orcid":"https://orcid.org/0009-0007-6299-2008","affiliations":[{"raw_affiliation_string":"Faculdade de Guarulhos (FG)","institution_ids":["https://openalex.org/I4210124990"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4210124990"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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/T12170","display_name":"History and Theory of Mathematics","score":0.39089998602867126,"subfield":{"id":"https://openalex.org/subfields/1207","display_name":"History and Philosophy of Science"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T12170","display_name":"History and Theory of Mathematics","score":0.39089998602867126,"subfield":{"id":"https://openalex.org/subfields/1207","display_name":"History and Philosophy of Science"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T14475","display_name":"History of Science and Medicine","score":0.23579999804496765,"subfield":{"id":"https://openalex.org/subfields/1207","display_name":"History and Philosophy of Science"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11960","display_name":"Relativity and Gravitational Theory","score":0.09220000356435776,"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/classical-mechanics","display_name":"classical mechanics","score":0.9639999866485596},{"id":"https://openalex.org/keywords/lagrangian-mechanics","display_name":"Lagrangian mechanics","score":0.968999981880188},{"id":"https://openalex.org/keywords/hamiltonian-mechanics","display_name":"Hamiltonian mechanics","score":0.9940000176429749},{"id":"https://openalex.org/keywords/hamilton-jacobi-equation","display_name":"Hamilton-Jacobi equation","score":0.8600000143051147},{"id":"https://openalex.org/keywords/variational-principle","display_name":"variational principle","score":0.9089999794960022},{"id":"https://openalex.org/keywords/noether-s-theorem","display_name":"Noether's theorem","score":0.7720000147819519},{"id":"https://openalex.org/keywords/conservation-laws","display_name":"conservation laws","score":0.8999999761581421},{"id":"https://openalex.org/keywords/symplectic-structure","display_name":"symplectic structure","score":0.7480000257492065},{"id":"https://openalex.org/keywords/canonical-structure","display_name":"canonical structure","score":0.7170000076293945},{"id":"https://openalex.org/keywords/phase-space","display_name":"phase space","score":0.6830000281333923},{"id":"https://openalex.org/keywords/history-of-mechanics","display_name":"history of mechanics","score":0.7509999871253967},{"id":"https://openalex.org/keywords/newton-s-principia","display_name":"Newton's Principia","score":0.8299999833106995}],"concepts":[{"id":"https://openalex.org/C80892491","wikidata":"https://www.wikidata.org/wiki/Q578555","display_name":"Noether's theorem","level":3,"score":0.8046000003814697},{"id":"https://openalex.org/C168619227","wikidata":"https://www.wikidata.org/wiki/Q2190991","display_name":"Symplectic geometry","level":2,"score":0.6909999847412109},{"id":"https://openalex.org/C499771625","wikidata":"https://www.wikidata.org/wiki/Q324669","display_name":"Lagrangian mechanics","level":5,"score":0.6416000127792358},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.6197999715805054},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5547999739646912},{"id":"https://openalex.org/C136864674","wikidata":"https://www.wikidata.org/wiki/Q477921","display_name":"Hamiltonian mechanics","level":3,"score":0.5342000126838684},{"id":"https://openalex.org/C206183056","wikidata":"https://www.wikidata.org/wiki/Q833065","display_name":"Analytical mechanics","level":4,"score":0.5289999842643738},{"id":"https://openalex.org/C140279962","wikidata":"https://www.wikidata.org/wiki/Q16002931","display_name":"Analytical dynamics","level":5,"score":0.515999972820282},{"id":"https://openalex.org/C121770821","wikidata":"https://www.wikidata.org/wiki/Q2072471","display_name":"Hamiltonian system","level":2,"score":0.47620001435279846},{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.46230000257492065},{"id":"https://openalex.org/C9259290","wikidata":"https://www.wikidata.org/wiki/Q745215","display_name":"Variational principle","level":2,"score":0.4163999855518341},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.41530001163482666},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.40400001406669617},{"id":"https://openalex.org/C52154503","wikidata":"https://www.wikidata.org/wiki/Q7915797","display_name":"Variational integrator","level":4,"score":0.39419999718666077},{"id":"https://openalex.org/C14037181","wikidata":"https://www.wikidata.org/wiki/Q215007","display_name":"Equations of motion","level":2,"score":0.382099986076355},{"id":"https://openalex.org/C28062496","wikidata":"https://www.wikidata.org/wiki/Q1145653","display_name":"Hamilton's principle","level":3,"score":0.3779999911785126},{"id":"https://openalex.org/C115071613","wikidata":"https://www.wikidata.org/wiki/Q1047080","display_name":"Differential form","level":2,"score":0.37209999561309814},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.3693000078201294},{"id":"https://openalex.org/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","level":2,"score":0.36079999804496765},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.359499990940094},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.35199999809265137},{"id":"https://openalex.org/C523394659","wikidata":"https://www.wikidata.org/wiki/Q17086892","display_name":"Variational analysis","level":2,"score":0.3513000011444092},{"id":"https://openalex.org/C64618476","wikidata":"https://www.wikidata.org/wiki/Q164296","display_name":"Variational method","level":2,"score":0.33410000801086426},{"id":"https://openalex.org/C4062443","wikidata":"https://www.wikidata.org/wiki/Q6701750","display_name":"Luke's variational principle","level":4,"score":0.31679999828338623},{"id":"https://openalex.org/C294558","wikidata":"https://www.wikidata.org/wiki/Q366968","display_name":"Newtonian fluid","level":2,"score":0.31450000405311584},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.3138999938964844},{"id":"https://openalex.org/C97985569","wikidata":"https://www.wikidata.org/wiki/Q216861","display_name":"Calculus of variations","level":2,"score":0.31349998712539673},{"id":"https://openalex.org/C53469067","wikidata":"https://www.wikidata.org/wiki/Q505735","display_name":"Lagrangian","level":2,"score":0.2955000102519989},{"id":"https://openalex.org/C138632511","wikidata":"https://www.wikidata.org/wiki/Q2478109","display_name":"Nonholonomic system","level":4,"score":0.29440000653266907},{"id":"https://openalex.org/C143218062","wikidata":"https://www.wikidata.org/wiki/Q722799","display_name":"Principle of least action","level":2,"score":0.29030001163482666},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.2892000079154968},{"id":"https://openalex.org/C111806380","wikidata":"https://www.wikidata.org/wiki/Q81745","display_name":"Classical physics","level":3,"score":0.27129998803138733},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.26499998569488525},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.25859999656677246},{"id":"https://openalex.org/C66783780","wikidata":"https://www.wikidata.org/wiki/Q1166625","display_name":"Mathematical problem","level":2,"score":0.25679999589920044},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25679999589920044},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.25220000743865967},{"id":"https://openalex.org/C33332235","wikidata":"https://www.wikidata.org/wiki/Q18362","display_name":"Theoretical physics","level":1,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.22209513","doi":"https://doi.org/10.5281/zenodo.22209513","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22209513","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"JournalArticle"},{"id":"doi:10.5281/zenodo.22209514","doi":"https://doi.org/10.5281/zenodo.22209514","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22209514","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"JournalArticle"}],"best_oa_location":{"id":"doi:10.5281/zenodo.22209513","doi":"https://doi.org/10.5281/zenodo.22209513","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22209513","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"JournalArticle"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2924432350","https://openalex.org/W3188783983","https://openalex.org/W1565865711","https://openalex.org/W7100383180","https://openalex.org/W4408345266","https://openalex.org/W2478381020","https://openalex.org/W3013347551","https://openalex.org/W2784030102","https://openalex.org/W4248307100","https://openalex.org/W4232400173"],"abstract_inverted_index":{"Classical":[0],"mechanics":[1],"is":[2,54,152],"physics'":[3],"longest-running":[4],"method":[5,102],"seminar:":[6],"the":[7,32,34,46,59,77,87,90,100,114,136,147,155,162,181,186,192,197,200],"same":[8],"motion":[9,165],"described":[10],"four":[11,173],"ways":[12],"---":[13,26,79,106,143,161,171],"Newton's":[14,64],"forces,":[15],"Lagrange's":[16,66],"variational":[17,82,97,101,123],"coordinates,":[18],"Hamilton's":[19,69],"phase-space":[20],"flow,":[21],"and":[22,73,86,108,113,119,128,131,139,190,196],"Hamilton-Jacobi's":[23],"complete":[24,88],"integral":[25],"each":[27],"formulation":[28],"an":[29],"instrument":[30,160],"for":[31,184,188,194,199],"problems":[33,98],"previous":[35],"could":[36],"not":[37,176],"see.":[38],"This":[39],"article":[40],"presents":[41],"a":[42,158,167,179],"narrative":[43],"review":[44],"of":[45,49,164,169],"primary":[47],"literature":[48,156],"those":[50],"formulations.":[51],"The":[52],"synthesis":[53],"organized":[55],"around":[56],"three":[57],"themes:":[58],"founding":[60],"methods,":[61],"in":[62,93,116],"which":[63,94,117],"Principia,":[65],"Mechanique":[67],"analytique,":[68],"two":[70],"dynamical":[71],"essays,":[72],"Jacobi's":[74],"lectures":[75],"built":[76],"ladder":[78],"differential":[80],"equations,":[81],"principles,":[83,124],"canonical":[84],"structure,":[85,195],"integral;":[89],"Noether":[91],"bridge,":[92],"Noether's":[95],"invariant":[96],"connected":[99],"to":[103],"conservation's":[104],"theorems":[105],"symmetry":[107],"conservation":[109],"as":[110,146,166],"one":[111],"statement;":[112],"consolidations,":[115],"Mach's":[118],"Dugas's":[120],"histories,":[121],"Lanczos's":[122],"Goldstein's":[125],"textbook,":[126],"Landau":[127],"Lifshitz's":[129],"Mechanics,":[130],"Arnold's":[132],"mathematical":[133],"methods":[134],"fixed":[135],"formulations'":[137],"pedagogy":[138],"their":[140],"modern":[141],"geometry":[142],"symplectic":[144],"structure":[145],"formulation's":[148],"common":[149],"tongue.":[150],"It":[151],"concluded":[153],"that":[154],"describes":[157],"single":[159],"description":[163],"choice":[168],"variables":[170],"whose":[172],"formulations":[174],"are":[175],"rivals":[177],"but":[178],"grammar:":[180],"Newtonian":[182],"equation":[183],"force,":[185],"Lagrangian":[187],"constraints":[189],"symmetry,":[191],"Hamiltonian":[193],"Hamilton-Jacobi":[198],"whole":[201],"trajectory":[202],"at":[203],"once.":[204]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-01T00:00:00"}