{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T07:15:54Z","timestamp":1783667754892,"version":"3.55.0"},"reference-count":46,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2027,1,1]],"date-time":"2027-01-01T00:00:00Z","timestamp":1798761600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2027,1,1]],"date-time":"2027-01-01T00:00:00Z","timestamp":1798761600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2027,1,1]],"date-time":"2027-01-01T00:00:00Z","timestamp":1798761600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2027,1,1]],"date-time":"2027-01-01T00:00:00Z","timestamp":1798761600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2027,1,1]],"date-time":"2027-01-01T00:00:00Z","timestamp":1798761600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2027,1,1]],"date-time":"2027-01-01T00:00:00Z","timestamp":1798761600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2027,1,1]],"date-time":"2027-01-01T00:00:00Z","timestamp":1798761600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12302001"],"award-info":[{"award-number":["12302001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12272248"],"award-info":[{"award-number":["12272248"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006247","name":"Anhui University of Science and Technology","doi-asserted-by":"publisher","award":["2023yjrc48"],"award-info":[{"award-number":["2023yjrc48"]}],"id":[{"id":"10.13039\/501100006247","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational and Applied Mathematics"],"published-print":{"date-parts":[[2027,1]]},"DOI":"10.1016\/j.cam.2026.117868","type":"journal-article","created":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T15:40:42Z","timestamp":1780501242000},"page":"117868","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Time-scale discrete Noether symmetries and structure-preserving algorithms for non-migratory Birkhoffian mechanics"],"prefix":"10.1016","volume":"489","author":[{"given":"Xue","family":"Tian","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kuan","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7703-1185","authenticated-orcid":false,"given":"Yi","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cam.2026.117868_bib0001","series-title":"Ein Ma\u00dfkettenkalk\u00fcl mit Anwendung auf Zentrumsmannigfaltigkeiten","author":"Hilger","year":"1988"},{"key":"10.1016\/j.cam.2026.117868_bib0002","series-title":"Dynamic Equations on Time Scales: An Introduction with Applications","author":"Bohner","year":"2001"},{"issue":"1\u20132","key":"10.1016\/j.cam.2026.117868_bib0003","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0377-0427(01)00432-0","article-title":"Dynamic equations on time scales: a survey","volume":"141","author":"Agarwal","year":"2002","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117868_bib0004","series-title":"Advances in Dynamic Equations on Time Scales","author":"Bohner","year":"2003"},{"key":"10.1016\/j.cam.2026.117868_bib0005","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/j.cam.2017.01.031","article-title":"Asymptotical properties of social network dynamics on time scales","volume":"319","author":"Ogulenko","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117868_bib0006","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2017.04.034","article-title":"On the exponential stability of Samuelson model on some classes of times scales","volume":"325","author":"Lizama","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117868_bib0007","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S0377-0427(01)00434-4","article-title":"Partial differential equations on time scales","volume":"141","author":"Ahlbrandt","year":"2002","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117868_bib0008","first-page":"339","article-title":"Calculus of variation on time scales","volume":"13","author":"Bohner","year":"2004","journal-title":"Dyn. Syst. Appl."},{"key":"10.1016\/j.cam.2026.117868_bib0009","series-title":"Multivariable Dynamic Calculus on Time Scales","author":"Bohner","year":"2016"},{"key":"10.1016\/j.cam.2026.117868_bib0010","series-title":"History of Analytical Mechanics","author":"Mei","year":"2019"},{"key":"10.1016\/j.cam.2026.117868_bib0011","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.cnsns.2015.11.020","article-title":"Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay","volume":"36","author":"Zhai","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.cam.2026.117868_bib0012","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.chaos.2018.12.005","article-title":"Noether\u2019s theorem of fractional Herglotz type in phase space","volume":"119","author":"Tian","year":"2019","journal-title":"Chaos Solitons Fract."},{"key":"10.1016\/j.cam.2026.117868_bib0013","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2023.107722","article-title":"Conservation laws for systems of non-standard Birkhoffian with fractional derivatives","volume":"130","author":"Zhang","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"2","key":"10.1016\/j.cam.2026.117868_bib0014","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1016\/j.jmaa.2008.01.018","article-title":"Noether\u2019s theorem on time scales","volume":"342","author":"Bartosiewicz","year":"2008","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"10.1016\/j.cam.2026.117868_bib0015","doi-asserted-by":"crossref","first-page":"9","DOI":"10.3166\/ejc.17.9-18","article-title":"The second Euler-Lagrange equation of variational calculus on time scales","volume":"17","author":"Bartosiewicz","year":"2011","journal-title":"Eur. J. Control"},{"issue":"1","key":"10.1016\/j.cam.2026.117868_bib0016","first-page":"87","article-title":"Noether\u2019s theorem for control problems on time scales","volume":"9","author":"Malinowska","year":"2014","journal-title":"Int. J. Differ. Equ."},{"issue":"1","key":"10.1016\/j.cam.2026.117868_bib0017","doi-asserted-by":"crossref","DOI":"10.1088\/1674-1056\/26\/1\/014501","article-title":"Methods of reduction for Lagrange systems on time scales with nabla derivatives","volume":"26","author":"Jin","year":"2017","journal-title":"Chin. Phys. B"},{"key":"10.1016\/j.cam.2026.117868_bib0018","first-page":"24","article-title":"Conserved quantities for Hamiltonian systems on time scales","volume":"313","author":"Song","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.cam.2026.117868_bib0019","doi-asserted-by":"crossref","DOI":"10.1063\/1.4932607","article-title":"Noether theorem for Birkhoffian systems on time scales","volume":"56","author":"Song","year":"2015","journal-title":"J. Math. Phys."},{"key":"10.1016\/j.cam.2026.117868_bib0020","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.cnsns.2017.04.012","article-title":"Noether theorem for non-conservative systems with time delay on time scales","volume":"52","author":"Zhai","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"5","key":"10.1016\/j.cam.2026.117868_bib0021","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1007\/s11433-013-5065-x","article-title":"Noether symmetries of the nonconservative and nonholonomic systems on time scales","volume":"56","author":"Cai","year":"2013","journal-title":"Sci. China Phys. Mech. Astron."},{"issue":"2","key":"10.1016\/j.cam.2026.117868_bib0022","doi-asserted-by":"crossref","DOI":"10.1088\/1674-1056\/27\/2\/020502","article-title":"Generalized Chaplygin equations for nonholonomic systems on time scales","volume":"27","author":"Jin","year":"2018","journal-title":"Chin. Phys. B"},{"issue":"8","key":"10.1016\/j.cam.2026.117868_bib0023","doi-asserted-by":"crossref","DOI":"10.1088\/1674-1056\/26\/8\/084501","article-title":"Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales","volume":"26","author":"Song","year":"2017","journal-title":"Chin. Phys. B"},{"key":"10.1016\/j.cam.2026.117868_bib0024","doi-asserted-by":"crossref","first-page":"4487","DOI":"10.1007\/s00707-022-03338-9","article-title":"Caputo-type fractional time-scales Noether theorem of Birkhoffian systems","volume":"233","author":"Tian","year":"2022","journal-title":"Acta Mech."},{"issue":"7","key":"10.1016\/j.cam.2026.117868_bib0025","first-page":"2010","article-title":"Caputo \u0394-type fractional time-scales Noether theorem","volume":"53","author":"Tian","year":"2021","journal-title":"Chin. J. Theor. Appl. Mech."},{"issue":"12","key":"10.1016\/j.cam.2026.117868_bib0026","article-title":"Noether\u2019s-type theorems on time scales","volume":"61","author":"Anerot","year":"2020","journal-title":"J. Math. Phys."},{"key":"10.1016\/j.cam.2026.117868_bib0027","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.cam.2016.02.047","article-title":"A new algorithm for solving dynamic equations on a time scale","volume":"312","author":"Jafari","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117868_bib0028","doi-asserted-by":"crossref","DOI":"10.1016\/j.taml.2022.100368","article-title":"A structure-preserving algorithm for time-scale non-shifted Hamiltonian systems","volume":"12","author":"Tian","year":"2022","journal-title":"Theor. Appl. Mech. Lett."},{"key":"10.1016\/j.cam.2026.117868_bib0029","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1017\/S096249290100006X","article-title":"Discrete mechanics and variational integrators","volume":"10","author":"Marsden","year":"2001","journal-title":"Acta Numer."},{"key":"10.1016\/j.cam.2026.117868_bib0030","series-title":"Symplectic Geometric Algorithms for Hamiltonian Systems","author":"Feng","year":"2003"},{"issue":"4-5","key":"10.1016\/j.cam.2026.117868_bib0031","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.physleta.2005.01.015","article-title":"Structure-preserving algorithms for Birkhoffian systems","volume":"336","author":"Sun","year":"2005","journal-title":"Phys. Lett. A"},{"issue":"1","key":"10.1016\/j.cam.2026.117868_bib0032","first-page":"1","article-title":"Structure-preserving algorithms for autonomous Birkhoffian systems","volume":"21","author":"Kong","year":"2012","journal-title":"J. Beijing Inst. Technol."},{"key":"10.1016\/j.cam.2026.117868_bib0033","article-title":"A multisymplectic Lie algebra variational integrator for flexible multibody dynamics on the special Euclidean group SE(3)","volume":"174","author":"Chen","year":"2022","journal-title":"Appl. Math. Mech."},{"issue":"5","key":"10.1016\/j.cam.2026.117868_bib0034","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1017\/S0956792521000139","article-title":"Structure-preserving deep learning","volume":"32","author":"Celledoni","year":"2021","journal-title":"Eur. J. Appl. Math."},{"key":"10.1016\/j.cam.2026.117868_bib0035","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1007\/s10915-023-02178-6","article-title":"Structure-preserving algorithms with uniform error bound and long-time energy conservation for highly oscillatory Hamiltonian systems","volume":"95","author":"Wang","year":"2023","journal-title":"J. Sci. Comput."},{"issue":"6","key":"10.1016\/j.cam.2026.117868_bib0036","first-page":"581","article-title":"A structure-preserving algorithm for Hamiltonian systems with nonholonomic constraints","volume":"41","author":"Man","year":"2020","journal-title":"Appl. Math. Mech."},{"issue":"5","key":"10.1016\/j.cam.2026.117868_bib0037","doi-asserted-by":"crossref","first-page":"1157","DOI":"10.1016\/j.geomphys.2011.12.004","article-title":"Structure-preserving algorithms for Birkhoffian systems","volume":"62","author":"Kong","year":"2012","journal-title":"J. Geom. Phys."},{"key":"10.1016\/j.cam.2026.117868_bib0038","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s13324-021-00594-1","article-title":"Two forms of the discrete equations and the Noether theorems for nonautonomous Birkhoffian systems","volume":"11","author":"Xia","year":"2021","journal-title":"Anal. Math. Phys."},{"key":"10.1016\/j.cam.2026.117868_bib0039","article-title":"Structure preserving algorithms with adaptive time step for Birkhoffian systems","volume":"480","author":"Kong","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.cam.2026.117868_bib0040","doi-asserted-by":"crossref","DOI":"10.1155\/2021\/9982975","article-title":"Time-scale version of generalized Birkhoffian mechanics and its symmetries and conserved quantities of Noether type","volume":"2021","author":"Chen","year":"2021","journal-title":"Adv. Math. Phys."},{"key":"10.1016\/j.cam.2026.117868_bib0041","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2021.106214","article-title":"Nonshifted dynamics of constrained systems on time scales under Lagrange framework and its Noether\u2019s theorem","volume":"108","author":"Zhang","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.cam.2026.117868_bib0042","doi-asserted-by":"crossref","DOI":"10.1016\/j.taml.2021.100286","article-title":"Mei\u2019s symmetry theorem for time scales nonshifted mechanical systems","volume":"11","author":"Zhang","year":"2021","journal-title":"Theor. Appl. Mech. Lett."},{"key":"10.1016\/j.cam.2026.117868_bib0043","doi-asserted-by":"crossref","DOI":"10.1155\/2021\/7329399","article-title":"Mei symmetry and conservation laws for time-scale nonshifted Hamilton equations","volume":"2021","author":"Zhang","year":"2021","journal-title":"Adv. Math. Phys."},{"issue":"24","key":"10.1016\/j.cam.2026.117868_bib0044","doi-asserted-by":"crossref","DOI":"10.7498\/aps.70.20210372","article-title":"Mei\u2019s symmetry theorems for non-migrated Birkhoffian systems on a time scale","volume":"70","author":"Zhang","year":"2021","journal-title":"Acta Phys. Sin."},{"key":"10.1016\/j.cam.2026.117868_bib0045","article-title":"Noether\u2019s theorems for nonshifted dynamic systems on time scales","volume":"374","author":"Song","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.cam.2026.117868_bib0046","series-title":"Dynamics of Birkhoffian Systems","author":"Mei","year":"1996"}],"container-title":["Journal of Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726005108?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726005108?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T06:43:17Z","timestamp":1783665797000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377042726005108"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2027,1]]},"references-count":46,"alternative-id":["S0377042726005108"],"URL":"https:\/\/doi.org\/10.1016\/j.cam.2026.117868","relation":{},"ISSN":["0377-0427"],"issn-type":[{"value":"0377-0427","type":"print"}],"subject":[],"published":{"date-parts":[[2027,1]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Time-scale discrete Noether symmetries and structure-preserving algorithms for non-migratory Birkhoffian mechanics","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational and Applied Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cam.2026.117868","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Published by Elsevier B.V.","name":"copyright","label":"Copyright"}],"article-number":"117868"}}