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[{"key": "dc.contributor.author", "value": "Brander, Tommi", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2012-04-26T07:49:25Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2012-04-26T07:49:25Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2012", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1215635", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/37724", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4 Pro gradu-tutkielma k\u00e4sittelee Hamiltonin ja Jacobin yht\u00e4l\u00f6it\u00e4, jotka kuvaavat mekaanisen j\u00e4rjestelm\u00e4n kehityst\u00e4 klassisen mekaniikan puitteissa. Hamiltonin ja Jacobin yht\u00e4l\u00f6it\u00e4 k\u00e4ytet\u00e4\u00e4n my\u00f6s s\u00e4\u00e4t\u00f6teoriassa sek\u00e4 kvanttimekaniikassa. Hamiltonin mekaaniikan kehitti Sir William Rowan Hamilton valon k\u00e4yt\u00f6ksen mallintamiseen ja Carl Gustav Jacob Jacobi kehitti sit\u00e4 edelleen.\n\nTutkielmassa annamme ehdot, joiden nojalla Hopfin ja Laxin kaava antaa ratkaisun Hamiltonin ja Jacobin yht\u00e4l\u00f6ihin liittyv\u00e4\u00e4n alkuarvo-ongelmaan. Sen j\u00e4lkeen m\u00e4\u00e4rit\u00e4mme sopivan heikon ratkaisun k\u00e4sitteen ja n\u00e4yt\u00e4mme heikkojen ratkaisujen olevan yksik\u00e4sitteisi\u00e4 tietyill\u00e4 ehdoilla. L\u00e4hestymme Hamiltonin ja Jacobin alkuarvo-ongelmaa asettamalla variaatio-ongelman, jonka Hopfin ja Laxin kaava ratkaisee. Osoitamme, ett\u00e4 Hopfin ja Laxin kaavan antama ratkaisuehdokas on Lipschitz-jatkuva ja toteuttaa dynaamisen ohjelmoinnin periaatteen, joka kytkee sen optimaalisen s\u00e4\u00e4d\u00f6n teoriaan. Sen j\u00e4lkeen n\u00e4yt\u00e4mme, ett\u00e4 Hopfin ja Laxin kaavan antama funktio todella ratkaisee Hamiltonin ja Jacobin yht\u00e4l\u00f6n alkuarvo-ongelman.\n\nT\u00e4rke\u00e4 ty\u00f6kalu Hopfin ja Laxin kaavan k\u00e4sittelyss\u00e4 on Legendren muunnos, joka muuntaa funktion sen konveksiksi duaaliksi. N\u00e4yt\u00e4mme, ett\u00e4 konvekseille ja tarpeeksi nopeasti kasvaville funktioille Legendren muunnos sovellettuna kahteen kertaan antaa alkuper\u00e4isen funktion takaisin. Tutkielmassa tutkitaan Hamiltonin ja Lagrangen funktioita, jotka t\u00e4ytt\u00e4v\u00e4t n\u00e4m\u00e4 ehdot.\n\nLopuksi m\u00e4\u00e4rittelemme, mit\u00e4 tarkoitamme heikolla ratkaisulla Hamiltonin ja Jacobin yht\u00e4l\u00f6n alkuarvo-ongelmaan. M\u00e4\u00e4ritelm\u00e4ss\u00e4 k\u00e4yt\u00e4mme semikonkaaveja funktioita. Osoitamme, ett\u00e4 alkuehtojen semikonkaavius tai Hamiltonin funktion vahva konveksisuus takaavat heikkojen ratkaisuiden semikonkaaviuden, ja ett\u00e4 semikonkaaveja ratkaisuja voi olla vain yksi, kunhan alkuarvo-ongelma t\u00e4ytt\u00e4\u00e4 sopivat s\u00e4\u00e4nn\u00f6llisyysehdot.", "language": null, "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Tommi Brander (toolbran) on 2012-04-26 07:49:23.465109. Form: Pro gradu -lomake (1 tekij\u00e4) (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake-1-tekijae). JyX data:", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija@noreply.fi) on 2012-04-26T07:49:24Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201204261578.pdf: 308714 bytes, checksum: f7c16816268f22f05092d6f86e23add3 (MD5)\nlicense.html: 4960 bytes, checksum: c0712ddfbe60f4d94d5ed716d77c6749 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2012-04-26T07:49:25Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201204261578.pdf: 308714 bytes, checksum: f7c16816268f22f05092d6f86e23add3 (MD5)\nlicense.html: 4960 bytes, checksum: c0712ddfbe60f4d94d5ed716d77c6749 (MD5)\n Previous issue date: 2012", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "27 s", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Hamilton-Jacobi equations", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201204261578", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.dcmitype", "value": "Text", "language": "en", "element": "type", "qualifier": "dcmitype", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2012-04-26T07:49:25Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "yht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "kaavat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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