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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Okkolin, Pauliina", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-08-17T05:59:59Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-08-17T05:59:59Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/77376", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4 tutkielma k\u00e4sittelee Brachistochrone-ongelmana tunnettavaa minimointiongelmaa. Ongelmassa on ideana l\u00f6yt\u00e4\u00e4 kahden tason pisteen A ja B v\u00e4linen k\u00e4yr\u00e4, joka minimoi ajan, joka massalliselta\nkappaleelta kuluu liukua pisteest\u00e4 A pisteeseen B.\nOngelma ratkaistaan t\u00e4ss\u00e4 ty\u00f6ss\u00e4 variaatiolaskentaa hy\u00f6dynt\u00e4en ja siten ty\u00f6 esittelee Brachistochrone-ongelman lis\u00e4ksi my\u00f6s tiettyj\u00e4 variaatiolaskennan\nperusideoita. Variaatiolaskenta on matemaattisen analyysin ala, joka tarjoaa keinoja \u00e4\u00e4riarvoteht\u00e4vien ratkaisemiseen, kun minimoitavat kuvaukset ovat\nfunktioavaruuksista reaaliluvuille m\u00e4\u00e4riteltyj\u00e4 funktionaaleja.\n\n\nTutkielmassa esitell\u00e4\u00e4n aluksi, kuinka sanallisesti muotoiltu ongelma saadaan johdettua matemaattiseen muotoon. Sen j\u00e4lkeen perehdyt\u00e4\u00e4n ongelman varsinaiseen ratkaisemiseen. Nyky\u00e4\u00e4n yleisesti tunnetaan, ett\u00e4 Brachistochrone-ongelman ratkaiseva k\u00e4yr\u00e4 on sykloidi. \nTy\u00f6ss\u00e4 n\u00e4ytet\u00e4\u00e4n, kuinka sykloidi variaatiolaskennan\navulla l\u00e4ht\u00f6kohtaisesti l\u00f6ydet\u00e4\u00e4n.\nKeskeisin ty\u00f6kalu on variaatiolaskennan \noleellisimpiin v\u00e4lineisiin kuuluva Euler-Lagrangen\ndifferentiaaliyht\u00e4l\u00f6. Ty\u00f6ss\u00e4 osoitetaan, ett\u00e4 Brachistochrone-ongelman ratkaisun on v\u00e4ltt\u00e4m\u00e4tt\u00e4 toteutettava Euler-Lagrangen yht\u00e4l\u00f6. Lis\u00e4ksi n\u00e4ytet\u00e4\u00e4n, ett\u00e4 jos Brachistochrone-ongelmalla on ratkaisu, se toteuttaa my\u00f6s Beltrami-yht\u00e4l\u00f6ksi kutsuttavan differentiaaliyht\u00e4l\u00f6n. Beltrami-yht\u00e4l\u00f6 ratkaisemalla saadaan n\u00e4ytetty\u00e4, ett\u00e4 ongelman mahdollinen ratkaisu on sykloidi.\n\n\nTy\u00f6n viimeinen vaihe on todistaa Brachistochrone-ongelman ratkaisun olemassaolo ja siten n\u00e4ytt\u00e4\u00e4, ett\u00e4 sykloidi todella ratkaisee ongelman.\nOlemassaolo todistetaan er\u00e4\u00e4n riitt\u00e4v\u00e4n ehdon avulla, joka kertoo, milloin Euler-Lagrangen yht\u00e4l\u00f6n toteuttava funktio on variaatio-ongelman ratkaisu. Ty\u00f6ss\u00e4 esitelt\u00e4v\u00e4 riitt\u00e4v\u00e4 ehto hy\u00f6dynt\u00e4\u00e4 funktioiden konveksisuutta. Riitt\u00e4v\u00e4 ehto ei ole suoraan sovellettavissa Brachistochrone-ongelmaan, joten ty\u00f6ss\u00e4 p\u00e4\u00e4dyt\u00e4\u00e4n viel\u00e4\ntarkastelemaan toista minimointiongelmaa, joka ratkaisemalla my\u00f6s Brachistochrone-ongelma saadaan ratkaistua.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-08-17T05:59:59Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-08-17T05:59:59Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "43", "language": "", "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": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "minimointiongelmat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "brakistokroni", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Euler-Lagrangen yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Brachistochrone-ongelma", "language": "", "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-202108174539", "language": "", "element": "identifier", "qualifier": "urn", "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": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"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": null, "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": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "variaatiolaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matemaattinen analyysi", "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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