Problem treh teles

delo diplomskega seminarja

Grega Saksida (Author), George Mejak (Mentor)

Abstract

Diplomsko delo z Lagrangeevo mehaniko razišče osnovne lastnosti restriktivnega problema treh teles. Najprej Lagrangeev izrek o vmesnih vrednostih odvedljive funkcije posplošimo na funkcije več spremenljivk. Z njim posplošimo Euler-Lagrangeevo enačbo variacijskega računa na funkcionale vektorskih funkcij. Nato je preko Hamiltonovega variacijskega principa uvedena Lagrangeeva mehanika. Definiramo Jacobijevo energijsko funkcijo in pokažemo, kdaj se ohranja. Uveden je pojem cikličnih koordinat in prikazano je, kako porodijo integrale gibanja, to so količine, ki se ohranjajo. Na nekaj osnovnih primerih je ponazorjeno delovanje Lagrangeeve mehanike. Sledi dokaz kovariantnosti Lagrangeeve mehanike. Naposled je z Lagrangeevo mehaniko opisan restriktivni problem treh teles. Zapišemo Lagrangeevo funkcijo za telo, katerega masa je zanemarljiva v primerjavi z masama drugih dveh teles. Sledijo izpeljave enačb, ki določajo ravnovesne lege tega telesa. Poiščemo obstoječe analitične rešitve. Za primer, ko je eno od masivnejših dveh teles mnogo masivnejše od drugega, v prvem redu določimo numerične približke ravnovesnih leg, ki se jih analitično ne da izraziti. Nazadnje je obravnavana še stabilnost najdenih ravnovesnih leg.

Keywords

Lagrangeva mehanika;problem treh teles;restriktivni problem treh teles;Lagrangeve točke;

Data

Language:	Slovenian
Year of publishing:	2020
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[G. Saksida]
UDC:	531/533
COBISS:	58571523
Views:	930
Downloads:	187
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The three-body problem
Secondary abstract:	The paper examines elementary properties of the restricted three-body problem. Lagrange's mean value theorem is generalised to multivariate case. Using this result, the Euler-Lagrange equations from the calculus of variations are generalised to functionals of vector functions. The Lagrangian mechanics is then derived from the Hamilton's principle. The Jacobi energy function is defined and its conservation properties examined. Cyclic coordinates are introduced and their connection to the integrals of motion is derived. The basic mechanisms of the Lagrangian mechanics are illustrated and explained by elementary examples. The covariance of the Lagrangian mechanics is proved. Finally, the restricted three-body problem is examined by the Lagrangian mechanics. The Lagrange function for the secondary (the body whose mass is negligible comparable to the masses of the other two bodies) is derived. Equations that determine the points of equilibrium are derived. Well known analytical solutions are presented. In the case where one of the primaries is of negligible mass comparable to the other one, solutions are numerically approximated to the first order. Finally, the stability of equilibrium points is examined.
Secondary keywords:	Lagrangian mechanics;three-body problem;restricted three-body problem;Lagrange points;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	40 str.
ID:	12027608