Publication
Direct and inverse variational problems on time scales: A survey
| Summary: | We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler–Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler–Lagrange equation (Helmholtz’s problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field. © 2017, Springer International Publishing AG. |
|---|---|
| Main Authors: | Dryl, M. |
| Other Authors: | Torres, D. F. M. |
| Subject: | Calculus of variations Dynamic equations on time scales Equation of variation Helmholtz’s problem Inverse problems Self-adjoint equations |
| Year: | 2017 |
| Country: | Portugal |
| Document type: | article |
| Access type: | restricted access |
| Associated institution: | Universidade de Aveiro |
| Language: | English |
| Origin: | RIA - Repositório Institucional da Universidade de Aveiro |
| _version_ | 1866172733447995392 |
|---|---|
| author | Dryl, M. |
| author2 | Torres, D. F. M. |
| author2_role | author |
| author_facet | Dryl, M. Torres, D. F. M. |
| author_role | author |
| country_str | PT |
| creators_json_txt | [{\"Person.name\":\"Dryl, M.\"},{\"Person.name\":\"Torres, D. F. M.\"}] |
| datacite.creators.creator.creatorName.fl_str_mv | Dryl, M. Torres, D. F. M. |
| datacite.date.Accepted.fl_str_mv | 2017-01-01T00:00:00Z |
| datacite.date.embargoed.fl_str_mv | 10000-01-01T00:00:00Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_16ec |
| datacite.subjects.subject.fl_str_mv | Calculus of variations Dynamic equations on time scales Equation of variation Helmholtz’s problem Inverse problems Self-adjoint equations |
| datacite.titles.title.fl_str_mv | Direct and inverse variational problems on time scales: A survey |
| dc.creator.none.fl_str_mv | Dryl, M. Torres, D. F. M. |
| dc.date.Accepted.fl_str_mv | 2017-01-01T00:00:00Z |
| dc.date.embargoed.fl_str_mv | 10000-01-01T00:00:00Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | http://hdl.handle.net/10773/18850 |
| dc.language.none.fl_str_mv | eng |
| dc.publisher.none.fl_str_mv | Springer |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_16ec |
| dc.subject.none.fl_str_mv | Calculus of variations Dynamic equations on time scales Equation of variation Helmholtz’s problem Inverse problems Self-adjoint equations |
| dc.title.fl_str_mv | Direct and inverse variational problems on time scales: A survey |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_6501 |
| description | We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler–Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler–Lagrange equation (Helmholtz’s problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field. © 2017, Springer International Publishing AG. |
| dirty | 0 |
| eu_rights_str_mv | restrictedAccess |
| format | article |
| id | ria_ee942d913a14ff407d9e2ee59bcf86ea |
| identifier.url.fl_str_mv | http://hdl.handle.net/10773/18850 |
| instacron_str | ua |
| institution | Universidade de Aveiro |
| instname_str | Universidade de Aveiro |
| language | eng |
| network_acronym_str | ria |
| network_name_str | RIA - Repositório Institucional da Universidade de Aveiro |
| oai_identifier_str | oai:ria.ua.pt:10773/18850 |
| organization_str_mv | urn:organizationAcronym:ua |
| person_str_mv | Dryl, M. Torres, D. F. M. |
| publishDate | 2017 |
| publisher.none.fl_str_mv | Springer |
| reponame_str | RIA - Repositório Institucional da Universidade de Aveiro |
| repository_id_str | urn:repositoryAcronym:ria |
| service_str_mv | urn:repositoryAcronym:ria |
| spelling | porWe deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler–Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler–Lagrange equation (Helmholtz’s problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field. © 2017, Springer International Publishing AG.application/pdfengSpringerporDirect and inverse variational problems on time scales: A surveyDryl, M.Torres, D. F. M.Handlehttp://hdl.handle.net/10773/18850ISSNIsPartOf2194-1009DOIIsPartOf10.1007/978-3-319-55236-1_1210000-01-01T00:00:00Z2017-01-01T00:00:00Z2017http://purl.org/coar/access_right/c_16ecrestricted accessporCalculus of variationsporDynamic equations on time scalesporEquation of variationporHelmholtz’s problemporInverse problemsporSelf-adjoint equations600160 byteshttp://purl.org/coar/access_right/c_16ecapplication/pdffulltexthttps://ria.ua.pt/bitstream/10773/18850/1/%5b338%5dDryl_Torres-survey.pdfliteraturehttp://purl.org/coar/resource_type/c_6501journal article |
| spellingShingle | Direct and inverse variational problems on time scales: A survey Dryl, M. Calculus of variations Dynamic equations on time scales Equation of variation Helmholtz’s problem Inverse problems Self-adjoint equations |
| status | SINGLETON |
| subject.fl_str_mv | Calculus of variations Dynamic equations on time scales Equation of variation Helmholtz’s problem Inverse problems Self-adjoint equations |
| title | Direct and inverse variational problems on time scales: A survey |
| title_full | Direct and inverse variational problems on time scales: A survey |
| title_fullStr | Direct and inverse variational problems on time scales: A survey |
| title_full_unstemmed | Direct and inverse variational problems on time scales: A survey |
| title_short | Direct and inverse variational problems on time scales: A survey |
| title_sort | Direct and inverse variational problems on time scales: A survey |
| topic | Calculus of variations Dynamic equations on time scales Equation of variation Helmholtz’s problem Inverse problems Self-adjoint equations |
| topic_facet | Calculus of variations Dynamic equations on time scales Equation of variation Helmholtz’s problem Inverse problems Self-adjoint equations |
| url | http://hdl.handle.net/10773/18850 |
| visible | 1 |