KlasiÃ„Âna mehanika

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SADR ˇ ZAJ ix 8.3.4 Kosi hitac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 8.3.5 Rijeke i cikloni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 8.4 Foucaultovo njihalo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 8.5 Općenita jednadˇzba gibanja čestice u neinercijskom sustavu . . . . . . . . . . . 290 9 Specijalna teorija relativnosti 293 9.1 Lorentzove transformacije . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 9.2 Relativistička kinematika . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 9.3 Relativistička dinamika . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 9.4 Hamiltonova formulacija relativističke mehanike . . . . . . . . . . . . . . . . . . 293 II Mehanika sustava čestica 295 10 Sustavi čestica 297 10.1 Diskretni i kontinuirani sustavi čestica . . . . . . . . . . . . . . . . . . . . . . . 297 10.2 Srediˇste mase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 10.3 Količina gibanja, moment količine gibanja, rad i energija: definicija i sačuvanje . 311 10.3.1 Količina gibanja sustava čestica . . . . . . . . . . . . . . . . . . . . . . . 311 10.3.2 Moment količine gibanja sustava čestica . . . . . . . . . . . . . . . . . . 314 10.3.3 Rad i energija sustava čestica . . . . . . . . . . . . . . . . . . . . . . . . 317 10.4 Neinercijski koordinatni sustav vezan za srediˇste mase . . . . . . . . . . . . . . . 322 10.5 Lagrangeovo i D’Alembertovo načelo . . . . . . . . . . . . . . . . . . . . . . . . 327 10.6 Sustavi s promjenjivom masom: gibanje rakete . . . . . . . . . . . . . . . . . . 336 10.7 Sudari čestica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 10.7.1 Centralni sudar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 10.7.2 Necentralni sudar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 11 Mali titraji sustava čestica 351 11.1 Mali longitudinalni titraji jednodimenzijskog diskretnog sustava čestica . . . . 352 11.1.1 Granica kontinuuma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 11.2 Mali transverzalni titraji kontinuiranog jednodimenzijskog sustava čestica . . . 364 11.2.1 Titranje napete niti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 11.2.2 Nit s oba nepomična ruba: stojni val (D. Bernoulli) . . . . . . . . . . . 369 11.2.3 Nit s oba nepomična ruba: putujući val (J. D’Alembert) . . . . . . . . 376 11.2.4 Nit s nepomičnim lijevim i slobodnim desnim rubom . . . . . . . . . . . 381 11.2.5 Nit sa slobodnim desnim i nepomičnim lijevim rubom . . . . . . . . . . 383 11.2.6 Nit slobodna na oba ruba . . . . . . . . . . . . . . . . . . . . . . . . . . 385 11.2.7 Brzina ˇsirenja grupe valova . . . . . . . . . . . . . . . . . . . . . . . . . . 387 11.2.8 Energija titranja napete niti . . . . . . . . . . . . . . . . . . . . . . . . . 387 11.3 Titranje pravokutne membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 11.4 Titranje kruˇzne membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 11.5 Titranje molekula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 11.5.1 Titranje simetrične linearne troatomske molekule . . . . . . . . . . . . . 400 11.5.2 Titranje molekule vode . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
x SADR ˇ ZAJ 12 Ravninsko gibanje krutog tijela 403 12.1 Uvod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 12.2 Moment tromosti krutog tijela . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 12.3 Teoremi o momentima tromosti . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 12.4 Parovi sila . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 12.5 Kinetička energija, rad i snaga vrtnje . . . . . . . . . . . . . . . . . . . . . . . . 414 12.6 Fizičko njihalo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 12.7 Općenito ravninsko gibanje krutog tijela . . . . . . . . . . . . . . . . . . . . . . 424 12.8 Trenutno srediˇste vrtnje . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 12.9 Statika krutog tijela . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 13 Prostorno gibanje krutog tijela 439 13.1 Tenzor tromosti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 13.2 Eulerove jednadˇzbe gibanja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 13.3 Gibanje Zemlje . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 13.4 Eulerovi kutovi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 13.5 Cayley - Klein parametri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 13.6 Gibanje zvrka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 III Analitička <strong>mehanika</strong> 473 14 Lagrangeove jednadˇzbe 475 14.1 Poopćene koordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 14.2 Stupnjevi slobode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 14.3 Lagrangeove jednadˇzbe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 14.4 Lagrangeove jednadˇzbe za impulsnu silu . . . . . . . . . . . . . . . . . . . . . . 494 14.5 Lagrangeova funkcija naelektrizirane čestice u elektromagnetskom polju . . . . 494 14.6 Hamiltonovo načelo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 14.7 Primjene Euler - Lagrangeove jednadˇzbe . . . . . . . . . . . . . . . . . . . . . . 500 14.8 Hamiltonovo načelo za neholonomne sustave . . . . . . . . . . . . . . . . . . . . 503 14.9 Funkcija djelovanja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 14.10Baˇzdarna preobrazba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 14.11Simetrije i zakoni sačuvanja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 15 Hamiltonove jednadˇzbe 509 15.1 Hamiltonove jednadˇzbe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 15.2 Poissonove zagrade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 15.3 Kanonska preobrazba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 15.3.1 Funkcija izvodnica kanonske preobrazbe . . . . . . . . . . . . . . . . . . 522 15.3.2 Infinitezimalna kanonska preobrazba . . . . . . . . . . . . . . . . . . . . 526 15.3.3 Poissonove zagrade i kanonska preobrazba . . . . . . . . . . . . . . . . . 528 15.4 Hamilton-Jacobijeva jednadˇzba . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 15.4.1 Fazni integrali - djelovanje i kutne varijable . . . . . . . . . . . . . . . . 539 15.5 Liouvilleov teorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 15.5.1 Kanonska preobrazba i Liouvilleov teorem . . . . . . . . . . . . . . . . . 545 15.6 Prijelaz na kvantnu mehaniku . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 15.6.1 Kanonski račub smetnje . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
Page 1 and 2: KLASIČNA MEHANIKA UVOD Zvonko Glum
Page 4 and 5: Sadrˇzaj 1 Uvod 1 I Mehanika jedne
Page 8 and 9: SADR ˇ ZAJ xi 16 Mali titraji sust
Page 10 and 11: Ovo su biljeˇske s autorovih preda
Page 12 and 13: Predgovor Iz područja teorijske me
Page 14 and 15: Kazalo kratica i simbola �A vekto
Page 16 and 17: Poglavlje 1 Uvod Na početku svake
Page 18 and 19: nego imaju i valna svojstva (kao ˇ
Page 20: Dio I Mehanika jedne čestice 5
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42 POGLAVLJE 2. MATEMATIČKI UVOD -
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92 POGLAVLJE 3. KINEMATIKA Slika 3.
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94 POGLAVLJE 3. KINEMATIKA Prvi čl
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96 POGLAVLJE 3. KINEMATIKA Slika 3.
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98 POGLAVLJE 3. KINEMATIKA Prva od
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100 POGLAVLJE 3. KINEMATIKA iz čeg
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102POGLAVLJE4. NEWTONOVIAKSIOMIGIBA
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126 POGLAVLJE 5. GIBANJE ČESTICE U
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152 POGLAVLJE 6. HARMONIJSKI OSCILA
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204 POGLAVLJE 7. GRAVITACIJA I CENT
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270 POGLAVLJE 8. INERCIJSKI I NEINE
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294 POGLAVLJE 9. SPECIJALNA TEORIJA
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Poglavlje 10 Sustavi čestica 10.1
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10.1. DISKRETNI I KONTINUIRANI SUST
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10.2. SREDI ˇ STE MASE 307 Slika 1
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10.2. SREDI ˇ STE MASE 309 Sa slik
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10.3. KOLI ČINAGIBANJA,MOMENTKOLI
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10.4. NEINERCIJSKI KOORDINATNI SUST
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10.4. NEINERCIJSKI KOORDINATNI SUST
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10.5. LAGRANGEOVO I D’ALEMBERTOVO
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10.6. SUSTAVI S PROMJENJIVOM MASOM:
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10.6. SUSTAVI S PROMJENJIVOM MASOM:
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10.7. SUDARI ČESTICA 341 Zadatak:
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10.7. SUDARI ČESTICA 343 R: Prema
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10.7. SUDARI ČESTICA 345 i poslije
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10.7. SUDARI ČESTICA 347 Slika 10.
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10.7. SUDARI ČESTICA 349 R: dovrˇ
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Poglavlje 11 Mali titraji sustava
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11.1. MALILONGITUDINALNITITRAJIJEDN
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11.2. MALITRANSVERZALNITITRAJIKONTI
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11.3. TITRANJE PRAVOKUTNE MEMBRANE
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11.4. TITRANJE KRU ˇ ZNE MEMBRANE
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11.5. TITRANJE MOLEKULA 401 Slika 1
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Poglavlje 12 Ravninsko gibanje krut
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12.2. MOMENT TROMOSTI KRUTOG TIJELA
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12.3. TEOREMI O MOMENTIMA TROMOSTI
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12.4. PAROVI SILA 413 izaziva trans
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12.5. KINETIČKA ENERGIJA, RAD I SN
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12.5. KINETIČKA ENERGIJA, RAD I SN
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12.6. FIZIČKO NJIHALO 419 Slika 12
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12.6. FIZIČKO NJIHALO 421 Izračun
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12.6. FIZIČKO NJIHALO 423 Uvrstiv
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12.7. OPĆENITO RAVNINSKO GIBANJE K
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12.8. TRENUTNO SREDI ˇ STE VRTNJE
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12.8. TRENUTNO SREDI ˇ STE VRTNJE
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12.9. STATIKA KRUTOG TIJELA 431 Uvj
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12.9. STATIKA KRUTOG TIJELA 433 Sad
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12.9. STATIKA KRUTOG TIJELA 435 Zad
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12.9. STATIKA KRUTOG TIJELA 437 Pro
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Poglavlje 13 Prostorno gibanje krut
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13.1. TENZOR TROMOSTI 441 komponent
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13.1. TENZOR TROMOSTI 443 njegove s
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13.1. TENZOR TROMOSTI 445 Uvrstimo
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13.1. TENZOR TROMOSTI 447 glavnih o
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13.2. EULEROVE JEDNAD ˇ ZBE GIBANJ
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13.2. EULEROVE JEDNAD ˇ ZBE GIBANJ
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13.3. GIBANJE ZEMLJE 453 Eulerove j
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13.3. GIBANJE ZEMLJE 455 (glavne os
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13.4. EULEROVI KUTOVI 457 13.4 Eule
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13.4. EULEROVI KUTOVI 459 Iz gornje
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13.5. CAYLEY - KLEIN PARAMETRI 461
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13.6. GIBANJE ZVRKA 463 sustav d
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13.6. GIBANJE ZVRKA 465 Integracijo
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13.6. GIBANJE ZVRKA 467 Slika 13.10
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13.6. GIBANJE ZVRKA 469 odeduju rub
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13.6. GIBANJE ZVRKA 471 Slika 13.13
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Dio III Analitička mehanika 473
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476 POGLAVLJE 14. LAGRANGEOVE JEDNA
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510 POGLAVLJE 15. HAMILTONOVE JEDNA
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554 POGLAVLJE 16. MALI TITRAJI SUST
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556 POGLAVLJE 16. MALI TITRAJI SUST
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558 POGLAVLJE 17. KLASIČNA TEORIJA
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Poglavlje 18 Mehanika Fluida uvodni
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Dodatak A Matematički dodatak A.1
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A.2. TAYLOROV RAZVOJ 583 A.2 Taylor
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A.4. ZAOKVIRENE FORMULE IZ OVE KNJI
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Dodatak B Diracova δ-funkcija Dira
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Sli”cnim se postupkom dobije i
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Na sli”can na”cin, u sfernom ko
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Dodatak C Presjeci stoˇsca: kruˇz
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Elipsa se definira kao ravninska kr
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Slika C.3: Uz izvod jednadˇzbe par
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Dodatak D Elementi Fourierove anali
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Izračunajmo sada integral IN ako u
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Dodatak E Vučedolski kalendar http
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već kao kanon stoji na tom prijelo
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Bibliografija [1] Aganović I., Ves
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Kazalo autora Bessel, FriedrichWilh
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Kazalo pojmova mehanicizam, 205 amp
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KAZALO POJMOVA 613 laplasijan, 46 L
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KAZALO POJMOVA 615 teorem Gaussov,
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