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Problem treh teles : delo diplomskega seminarja
ID Saksida, Grega (Author), ID Mejak, George (Mentor) More about this mentor... This link opens in a new window

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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.

Language:Slovenian
Keywords:Lagrangeeva mehanika, problem treh teles, restriktivni problem treh teles, Lagrangeeve točke
Work type:Bachelor thesis/paper
Organization:FMF - Faculty of Mathematics and Physics
Year:2020
PID:20.500.12556/RUL-119286 This link opens in a new window
UDC:531/533
COBISS.SI-ID:58571523 This link opens in a new window
Publication date in RUL:06.09.2020
Views:802
Downloads:181
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Secondary language

Language:English
Title:The three-body problem
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.

Keywords:Lagrangian mechanics, three-body problem, restricted three-body problem, Lagrange points

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