{"id":"https://openalex.org/W7133482091","doi":"https://doi.org/10.3390/e28030292","title":"From Dirac Structures to Port-Hamiltonian Partial Differential Equations, a Tutorial Introduction","display_name":"From Dirac Structures to Port-Hamiltonian Partial Differential Equations, a Tutorial Introduction","publication_year":2026,"publication_date":"2026-03-04","ids":{"openalex":"https://openalex.org/W7133482091","doi":"https://doi.org/10.3390/e28030292","pmid":"https://pubmed.ncbi.nlm.nih.gov/41899944"},"language":"en","primary_location":{"id":"doi:10.3390/e28030292","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030292","pdf_url":"https://www.mdpi.com/1099-4300/28/3/292/pdf?version=1772703329","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/28/3/292/pdf?version=1772703329","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076905620","display_name":"H. J. Zwart","orcid":null},"institutions":[{"id":"https://openalex.org/I83019370","display_name":"Eindhoven University of Technology","ror":"https://ror.org/02c2kyt77","country_code":"NL","type":"education","lineage":["https://openalex.org/I83019370"]},{"id":"https://openalex.org/I94624287","display_name":"University of Twente","ror":"https://ror.org/006hf6230","country_code":"NL","type":"education","lineage":["https://openalex.org/I94624287"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Hans Zwart","raw_affiliation_strings":["Department of Applied mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands","Faculty of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands"],"raw_orcid":"https://orcid.org/0000-0003-3451-7967","affiliations":[{"raw_affiliation_string":"Department of Applied mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands","institution_ids":["https://openalex.org/I94624287"]},{"raw_affiliation_string":"Faculty of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5076905620"],"corresponding_institution_ids":["https://openalex.org/I83019370","https://openalex.org/I94624287"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.22284198,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"28","issue":"3","first_page":"292","last_page":"292"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.9186000227928162,"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"}},"topics":[{"id":"https://openalex.org/T12772","display_name":"Control and Stability of Dynamical Systems","score":0.9186000227928162,"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/T11262","display_name":"Quantum Mechanics and Non-Hermitian Physics","score":0.012000000104308128,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.009200000204145908,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/dirac","display_name":"Dirac (video compression format)","score":0.7803999781608582},{"id":"https://openalex.org/keywords/dirac-measure","display_name":"Dirac measure","score":0.5430999994277954},{"id":"https://openalex.org/keywords/dirac-spinor","display_name":"Dirac spinor","score":0.5353999733924866},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.532800018787384},{"id":"https://openalex.org/keywords/dirac-algebra","display_name":"Dirac algebra","score":0.5295000076293945},{"id":"https://openalex.org/keywords/differential","display_name":"Differential (mechanical device)","score":0.5013999938964844},{"id":"https://openalex.org/keywords/dirac-operator","display_name":"Dirac operator","score":0.4496999979019165},{"id":"https://openalex.org/keywords/dirac-comb","display_name":"Dirac comb","score":0.4327000081539154}],"concepts":[{"id":"https://openalex.org/C61039578","wikidata":"https://www.wikidata.org/wiki/Q604279","display_name":"Dirac (video compression format)","level":3,"score":0.7803999781608582},{"id":"https://openalex.org/C111118313","wikidata":"https://www.wikidata.org/wiki/Q1227387","display_name":"Dirac measure","level":5,"score":0.5430999994277954},{"id":"https://openalex.org/C156785651","wikidata":"https://www.wikidata.org/wiki/Q688945","display_name":"Dirac spinor","level":4,"score":0.5353999733924866},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.532800018787384},{"id":"https://openalex.org/C149545384","wikidata":"https://www.wikidata.org/wiki/Q5280160","display_name":"Dirac algebra","level":3,"score":0.5295000076293945},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.5013999938964844},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4677000045776367},{"id":"https://openalex.org/C158650840","wikidata":"https://www.wikidata.org/wiki/Q907439","display_name":"Dirac operator","level":2,"score":0.4496999979019165},{"id":"https://openalex.org/C50953366","wikidata":"https://www.wikidata.org/wiki/Q385599","display_name":"Dirac comb","level":5,"score":0.4327000081539154},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3864000141620636},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.38440001010894775},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3467999994754791},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.34439998865127563},{"id":"https://openalex.org/C2984217123","wikidata":"https://www.wikidata.org/wiki/Q25099949","display_name":"Structure function","level":2,"score":0.32659998536109924},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3228999972343445},{"id":"https://openalex.org/C50575695","wikidata":"https://www.wikidata.org/wiki/Q1227393","display_name":"Dirac sea","level":4,"score":0.2842000126838684},{"id":"https://openalex.org/C82601208","wikidata":"https://www.wikidata.org/wiki/Q272621","display_name":"Dirac equation","level":2,"score":0.28299999237060547},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.2741999924182892},{"id":"https://openalex.org/C33332235","wikidata":"https://www.wikidata.org/wiki/Q18362","display_name":"Theoretical physics","level":1,"score":0.2578999996185303},{"id":"https://openalex.org/C54486999","wikidata":"https://www.wikidata.org/wiki/Q209675","display_name":"Dirac delta function","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e28030292","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030292","pdf_url":"https://www.mdpi.com/1099-4300/28/3/292/pdf?version=1772703329","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:41899944","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/41899944","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:ris.utwente.nl:openaire/aa38b906-8237-4f81-bb80-578e9c659d7d","is_oa":true,"landing_page_url":"https://research.utwente.nl/en/publications/aa38b906-8237-4f81-bb80-578e9c659d7d","pdf_url":null,"source":{"id":"https://openalex.org/S4406922991","display_name":"University of Twente Research Information","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Zwart, H 2026, 'From Dirac Structures to Port-Hamiltonian Partial Differential Equations, a Tutorial Introduction', Entropy, vol. 28, no. 3, 292. https://doi.org/10.3390/e28030292","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:f9f787836f27451ba3aca0d46c7c31b4","is_oa":true,"landing_page_url":"https://doaj.org/article/f9f787836f27451ba3aca0d46c7c31b4","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 28, Iss 3, p 292 (2026)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:13025555","is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC13025555/","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e28030292","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030292","pdf_url":"https://www.mdpi.com/1099-4300/28/3/292/pdf?version=1772703329","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7133482091.pdf","grobid_xml":"https://content.openalex.org/works/W7133482091.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,50,107],"discuss":[4],"the":[5,28,55,64,87,97,110,115,121,135],"geometric":[6],"structure,":[7,10,86],"i.e.,":[8,81],"Dirac":[9,33,59,85,116,130],"underlying":[11],"port-Hamiltonian":[12],"systems.":[13],"The":[14,129],"paper":[15],"has":[16,139],"a":[17,32,43,58],"tutorial":[18],"character,":[19],"and":[20,35],"thus":[21],"it":[22],"contains":[23],"questions/exercises.":[24],"We":[25,94],"start":[26],"with":[27,103],"general":[29],"definition":[30],"of":[31,66,112,127],"structure":[34,60,117,131],"show":[36,51,95],"that":[37,96,134],"on":[38,82],"finite-dimensional":[39,56],"spaces,":[40],"there":[41],"is":[42],"simple":[44,48],"matrix":[45],"characterization.":[46],"By":[47],"examples,":[49],"that,":[52],"even":[53,91],"in":[54],"case,":[57],"does":[61],"not":[62],"guarantee":[63],"existence":[65,88,111],"solutions":[67],"for":[68],"an":[69,83],"associated":[70,77,136],"ordinary":[71],"differential":[72,79],"or":[73],"difference":[74],"equation.":[75],"For":[76],"partial":[78],"equations,":[80],"infinite-dimensional":[84],"problem":[89],"becomes":[90],"more":[92],"challenging.":[93],"spaces":[98],"have":[99,108],"to":[100],"be":[101],"chosen":[102],"care,":[104],"but":[105],"when":[106],"shown":[109],"solutions,":[113],"then":[114],"will":[118],"give":[119],"us":[120],"desired":[122],"properties,":[123],"such":[124],"as":[125],"conservation":[126],"energy.":[128],"also":[132],"implies":[133],"transfer":[137],"function":[138],"nice":[140],"properties.":[141]},"counts_by_year":[],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2026-03-05T00:00:00"}
