Detaljna analiza problema brahistokrone

master's thesis

• Author: Ana Radić
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Sonja Štimac (član povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Problem koji se obrađuje u ovom radu je pronalazak puta kojim bi najbrže stigli iz jedne u drugu točku uz utjecaj jedino sile teže. Poznatiji je pod imenom Brahistokrona krivulja te ga je popularizirao švicarski matematičar Johann Bernoulli 1696 godine, koji je godinu kasnije i arugmentirao da je rješenje cikloida. U radu smo problemu pristupili na tri načina: varijacijskim računom, optimalnim upravljanjem i analitčkim rješavanjem baziranim na Cauchy-Schwarzovoj nejednakosti. Cilj je za opću krivulju doći do izraza za ukupno vrijeme kretanja, a potom ga minimizirati, odnosno pronaći krivulju za koju se minimum postiže. Prvi korak je odrediti sve moguće kandidate za minimum koje dobivamo evaluiranjem nužnih uvjeta optimalnosti. Preciznije, u teoriji varijacijskog računa riječ je o Euler-Lagrangeovim jednadžbama, a u optimalnom upravljanju koristimo Pontrjaginov princip maksimuma. U oba slučaja dobivamo da je cikloida jedini kandidat za rješenje. Koristeći konveksnost teorijom varijacijskog računa dobivamo da je cikloida zaista jedinstvena Brahistokrona. Istaknimo još i da singularitet i nekonveksnost podintegralne funkcije znatno otežavaju ovu analizu, što je u ovom radu detaljno i raspisano.

Abstract (english)

In this thesis, we present the problem of finding the path along which an object would slide in the shortest possible time from point A to point B if it is only accelerated by gravity. This problem, known as the Brachistochrone curve problem, was first published by Johann Bernoulli in 1696, who showed that its solution is cycloid. This work contains different methods to get the equation of the Brachistochrone curve, precisely calculus of variations approach, optimal control, and the one that uses only Cauchy-Schwarz inequality. Considering the goal is to find the shortest time, the basic idea is to minimize the functional that describes the total travel time from one point to another to get a time-optimal solution. The first step is to find all the candidates for a minimum by finding sufficient and necessary conditions for the extrema of the functional. In the calculus of variations method, we do it by using Euler Lagrange equations, and for the optimal control method, we use Pontrjagin maximum principle. In both cases, we get that the cycloid is a unique solution candidate. In the first method, we use convexity to prove that it is the unique solution. Also, singularity and nonconvexity of an integrand make this analysis harder.

• Keywords: Brahistokrona krivulja; Johann Bernoulli; varijacijski račun; optimalno upravljanje; Cauchy-Schwarzovoj nejednakost
• Keywords (english): Brachistochrone curve; Johann Bernoulli; variations approach; optimal control; Cauchy-Schwarz inequality
• Language: croatian
• URN:NBN: urn:nbn:hr:217:432766
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-11-30 11:49:01