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undergraduate thesis
Problem brahistokrone u nehomogenom gravitacijskom polju
Split: University of Split, Faculty of Science, 2022. urn:nbn:hr:166:949237
Šalinović, Josip
University of Split
Faculty of Science
Department of Physics

Institutional repository: Repository of Faculty of Science

Cite this document

Šalinović, J. (2022). Problem brahistokrone u nehomogenom gravitacijskom polju (Undergraduate thesis). Split: University of Split, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:166:949237

Šalinović, Josip. "Problem brahistokrone u nehomogenom gravitacijskom polju." Undergraduate thesis, University of Split, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:166:949237

Šalinović, Josip. "Problem brahistokrone u nehomogenom gravitacijskom polju." Undergraduate thesis, University of Split, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:166:949237

Šalinović, J. (2022). 'Problem brahistokrone u nehomogenom gravitacijskom polju', Undergraduate thesis, University of Split, Faculty of Science, accessed 17 July 2024, https://urn.nsk.hr/urn:nbn:hr:166:949237

Šalinović J. Problem brahistokrone u nehomogenom gravitacijskom polju [Undergraduate thesis]. Split: University of Split, Faculty of Science; 2022 [cited 2024 July 17] Available at: https://urn.nsk.hr/urn:nbn:hr:166:949237

J. Šalinović, "Problem brahistokrone u nehomogenom gravitacijskom polju", Undergraduate thesis, University of Split, Faculty of Science, Split, 2022. Available at: https://urn.nsk.hr/urn:nbn:hr:166:949237

Metadata
TitleProblem brahistokrone u nehomogenom gravitacijskom polju
Title (english)Brachistochrone problem for nonuniform gravitational field
AuthorJosip Šalinović
MentorPetar Stipanović (mentor)
Committee memberPetar Stipanović (predsjednik povjerenstva)
Committee memberŽeljana Bonačić Lošić (član povjerenstva)
Committee memberMarko Kovač (član povjerenstva)
GranterUniversity of Split
Faculty of Science
(Department of Physics)
Split
Defense date and country2022-09-27, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Physics
Abstract
Problem brahistokrone jedan je od zanimljivijih problema klasične mehanike koji potječe još od vremena Galilea. Problem je glasio ovako: Dane su točke A i B koje su položene u vertikalnoj ravnini, tijelo pod utjecajem gravitacije klizi po krivulji bez trenja od točke A do B. Nađite krivulju takvu da tijelo u najkraćem mogućem vremenu stigne od A do B. Prva točna rješenja objavili su braća Bernoulli koja su koristila zakone optike, točnije Snellov zakon kao polazište. Kasnije se razvila nova grana matematike varijacijski račun, te Euler-Lagrangeova jednadžba koja je problem brahistokrone riješila analitičkim putem. U ovom radu prezentiramo analitičko rješenje problema brahistokrone iz Euler-Lagrangeove jednadžbe i numeričko rješenje pripadne diferencijalne jednadžbe. Uspoređujemo rješenja numeričke metode s analitičkim rješenjima. Na kraju je problem brahistokrone generaliziran za nehomogena gravitacijska polja.
Abstract (english)
The problem of brachistochrone is one of the most interesting problems of classical mechanics dating back to the time of Galileo. The problem was as follows: Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time. Johann and his brother Jakob Bernoulli derived the first solution. They used the laws of optics more precisely Snell's law as indirect method. Later, a new branch of the mathematics of variational calculus was developed and the Euler-Lagrange equation solved the problem of brachistochrone analytically. In this paper, we will present an analytical solution of the brachistochrone problem from the Euler-Lagrange equation and a numerical solution of the corresponding differential equation. Finally, we will generalize the brachistochrone problem for nonuniform gravitational fields.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:166:949237
Study programmeTitle: Physics
Study programme type: university
Study level: undergraduate
Academic / professional title: sveučilišni/a prvostupnik/prvostupnica (baccalaureus/baccalaurea) fizike (sveučilišni/a prvostupnik/prvostupnica (baccalaureus/baccalaurea) fizike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence Attribution 4.0 International (CC BY 4.0) icon
Created on2023-03-01 11:12:06