Problem globalnosti akcijsko-kotnih koordinat : magistrsko delo
Novak, Tina, 1977-
V nalogi dokažemo Arnold-Liouvillov izrek za Liouvillov integrabilni sistem. Za tak sistem obstajajo semi-globalne kanonične akcijsko-kotne koordinate, rešitev sistema pa se da zapisati z integrali. ...Za splošnejšo situacijo, sveženj ▫$ M^{2n} \to B^n$▫, pri čemer je ▫$*M,\omega)$▫ simplektična mnogoterost, vlakna difeomorfna torusu ▫$T^n$▫, sveženjska projekcija pa regularna preslikava na ▫$n$▫-mnogoterost ▫$B$▫, definiramo monodromijo in Chern-Duistermaatov razred. To sta obstrukciji za obstoj globalnih akcijsko-kotnih koordinat. Za harmonični oscilator in za sistem ▫$n$▫ neodvisnih oscilatorjev izračunamo akcijsko-kotne koordinate. Za matematično nihalo pa akcijsko koordinato zapišemo v kvadraturah. Zanimiv primer Hamiltonovega sistema je sferično nihalo. Zaradi simetrije sistema znamo poiskati prvi integral, to je kotni moment. Pokažemo, da za sistem sferičnega nihala ne obstajajo globalne akcijsko-kotne koordinate, saj ima netrivialno monodromijo. Chern-Duistermaatov razred pa je trivialen.
Type of material - master's thesis
; adult, serious
Publication and manufacture - Ljubljana : [T. Novak], 2006
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