20.500.12556/RUL-119286 Problem treh teles delo diplomskega seminarja The three-body problem 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. 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. Lagrangeeva mehanika problem treh teles restriktivni problem treh teles Lagrangeeve točke Lagrangian mechanics three-body problem restricted three-body problem Lagrange points true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2020-09-06 08:15:50 2020-09-06 08:15:58 2024-05-30 09:24:39 0000-00-00 00:00:00 2020 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 531/533 108414 58571523 1394.pdf 1394.pdf 1 2466483310D357E79603E8B86A66F7EA 117d1a1e3352925eb3cdd06d4983dd0367ffdc2ecf038305878bcde591aec347 c99b2bd5-a1b9-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=133984 Fakulteta za matematiko in fiziko 0 0 0