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U ia r n in g ik k i n c h i tartibli hosilalari
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U ia r n in g ik k i n c h i tartibli hosilalari: Jt = - 1 ф 2 sin -l cos ( р - 1 ф sin (p. (1 .1 0 9 ) Bularni b ir in c h i ikkita ten g la m a la r g a q o 'y a m iz : m l ( ф cos (p- ф 2 sin (p = 0; m l (ф ы п (р + ф 2 c o s c p j - 2 X l c o s c p = - m g . Ikkita n o m a ’lu m u c h u n ikkita te n g l a m a qold i. U ia r n in g b irinchis in i c o s ф ga v a ik kin ch is in i sin сp ga ko'p aytir ib, qo's h ilsa Ф u c h u n t e n g l a m a hosil bo'ladi: (p + y s i n < p = 0. ( 1 . 1 1 1 ) H o s il b o 'lg a n te n g l a m a (1 .1 0 5 ) n in g ay n a n o'zi dir. Lagranj k o 'p a y t u v c h isi X ni h a m to p i s h oson: . 1 1 A = - m ( p ~ + — m g c o s (p. ( 1 . 1 1 2 ) 31 K o ' r in i b turibdiki, X m u s ta q il o ‘z g a r u v c h i e m a s , u n i n g q iy m a ti t o ' li q ravis hda ( 1 . 1 1 1 ) n in g y e c h i m i (p(t) orqali a n iq l a n a d i. Lagranj k o ‘p aytu vch isi X d i n a m i k o ‘zgaruvchi b o i m a s a h a m u q a n d a y d ir o ‘z g a r u v c h i, u m u m l a s h g a n k o o r d i n a t a l a r n i n g biri. U n i n g m a n a s h u x u s u s i y m i s o l d a g i m a ’n o s i n i m a d a n iborat? Oxirgi fo r m u la d a n k o ‘rinib tu ribdik i 2 Л/ m a y a t n ik k a t a ’sir q i l a y o t g a n ik kita k u c h n i n g y i g ' i n d i s i g a t e n g : m a r k a z d a n q o c h m a k u c h p lu s gravitats iya k u c h i n i n g ip g a p r o y e k siy a si. H a q i q a t a n h a m , ( 1 . 9 8 ) d an k o'rin ib turibdiki, h ad b o g i a n i s h or q a li p a y d o b o ‘lg an k u c h m a ’n o s i g a 1.6.4 .-m iso l. Y e r b i l a n (p b u r c h a k h o s i l q i l g a n qiyalik b o ' y i c h a harakat q i l a y o tg a n a radiusli s ilin d r m asalasin i qarab c h iq a y l ik . M a s a l a d a i k k i t a u m u m l a s h g a n k o o r d i n a t a b o r q = x ,q ~ = 9 . S il i n d r i m i z s i r p a n m a s d a n tu s h a y a p ti d e - m o q c h i b o 'l s a k u n i n g harakatini а в = х d - И З ) shartga b o 'y s u n d ir i s h kerak. B u shart silin d r n in g t e k i s - likka tegib tu rg an n u q t a s i n i n g ilgarilanm a harakat tezlig i u n i n g a y l a n m a h a r a k a t t e z l i g i g a t e n g l i g i n i b i l d i r a d i . Y a ’n i , b u s h a r t i s h q a l a n i s h k u c h i b o r l i g i n i h i s o b g a o li s h g a tengdir. Shart o ‘z k o 'r in is h i b o ' y i c h a n o g o l o n o m b o 'l is h ig a q a ra m a y u n i in t e g r a ll a b tez lik la r k i r m a y d i g a n , fa q a t u m u m l a s h g a n k o o r d in a t la r g a b o g ' l i q b o 'i g a n shaklga keltirishi m u m k in : d ( а в - x ) = 0 —> а в - x = const. ( 1 . 1 1 4 ) S il in d r n i n g kin etik en erg iy a s i ikki q i s m d a n iborat: T - ^ m x ' + ^ т к ' в ‘ . ( 1 . 1 1 5 ) B ir in c h i had il garilanma harakat kin etik e n e r g iy a s i, ik kinchi had a y la n m a harakat kin etik energ iy asi («Q attiq j i s m » b o b ig a qarang). ik k i n c h i haddagi k o e f f i t s i v e n t s i li n d r n i n g in e r s iy a m o m e n t i b il a n a n i q l a n a d i , u n i n g a n iq t a ’rifi h o z ir zarur em as. P o t e n s i a l e n e r g iy a n i U = m£(/-.Y)sin (1.116) k o 'r i n is h d a o l a m i z B u n d a / - q iy a l ik n i n g u z u n li g i. m g l sin s i lin d r n in g harakat boshidagi p o te n s ia l e n e r g iy a s i. A lb atta, m g l sin cp o 'z g a r m a s h a d n i ta shla b yuborish m u m k i n , u harakat ten g la m a la r ig a t a ’sir q il m a y d i , a m m o un i p o te n s i a l i n i n g fi zik m a ' n o s i n i y o r it is h g a x iz m a t qilgan t u c h u n q o id i r a m i z . D e m a k , Lagranj fu n k siyasi e g a b o 'l is h i kerak. 1.2- rasm . Q iyalik b o 'y ich a tu sh ayotgan silindr. 32 2 2 H a ra k a t t e n g l a m a la r i g a o ‘ta ylik . B u h o i s h u n d a y h o l k i , g o l o n o m va n o g o l o n o m hollardagi m et odla rnin g ikkalasini h a m q o i l a s h i m u m k in , ch u n k i bog'lanis hni ikkala k o ‘rinishda yo z ib o lin d i — (1 .1 1 3 ) va (1 .1 1 4 ) formulalar. A gar g o l o n o m h o li d a g i m e t o d d a n f o y d a l a n il s a , Lagranj k o ‘p a y t u v c h i s i n i bevosita Lagranj fu nksiyasiga kiritiladi: bu n d a c) ixtiyoriy konst anta . U c h t a o ‘zgaruvchi bor x, в, A. M a n a sh u u m u m la s h g a n koordin atla r b o ' y i c h a harakat tenglam ala ri y o z ib olinadi: Oxirgi te n g la m a n i а в = x ko'r inis hga keltirib olib, o d d iy h is o b y o r d a m i d a ek a n lig in i to p a m iz . A ning m a ’n o s i n i — F - dL '/d x = m g sin la dan olish m u m k in . F u m u m la s h g a n k uch, u ikki q is m d a n iborat: birin chisi tortish k u ch i, ik kinchis i ishqala nish kuchi. D e m a k , qiyalik b o 'y i c h a ayla nib t u s h a y o tg a n silindrga t a ’sir qilayotgan is h q a la n ish k u c h in i to p d ik . S i l i n d r n i n g t e z l a n i s h i g a k e la y l ik . A g a r q iy a l ik n i a b s o l u t t e k i s y a ’ni, m a s a la d a h e c h q a n d a y is h q a la n i s h y o ' q d e b o ls a k , ( 1 . 1 1 3 ) shart p a y d o b o 'l m a s ed i, silindr ay la nib tushishi kerak h a m b o 'l m a s edi. Y a ’ni, kinetik en erg iy a ha did a 0 ga bog'liq lik ham p a y d o b o ' l m a s ed i. Bu h o id a (1 .1 2 0 ) va ( 1 . 1 2 1 ) t e n g l a m a l a r d a к = 0 d e b o l i s h kerak (b u b ila n s i l i n d r n i n g a y la n m a harakatini ch iq arib ta s h la y m i z ) . K o'rin ib tu ribdik i. i s h q a l a n i s h n i n g m a v j u d lig i si li n d r n in g t e z l a n is h i n i k am aytir adi. H a ra k a t t e n g l a m a l a r i n i i n t e g r a ll a s h q i y i n e m a s . Q i y a l i k n i n g u c h i d a turgan silindrn in g b o s h l a n g 'i c h te zligin i nol d eb quyid agi topiladi: L ' = L + X ( a Q — x — C \ ) , ( 1 . 1 1 8 ) m k 0 = a X : а в - x = с , . mx = m g sin cp - Я; ( 1 . 1 1 9 ) ( 1 . 120 ) va a g x = —5 ------ r- s in a + k ~ ( 1. 121) 3 — Nazariy mexanika 33 1 ' ' 1 a gt~ . , . х - ■■■? - - y sin ( 1 . 1 2 2 ) 2 сГ +к~ Bu f o r m u la d a n s i li n d r n in g q iy a lik n in g o x ir i g a c h a y e t i b b o r is h i u c h u n q a n c h a v a q t kerak d e g a n s a v o l g a j a v o b t o p i s h q i y i n e m a s , b u n i n g u c h u n x ( t n) = l t e n g l a m a n i y e c h i s h kerak xolo s . Shart b o s h i d a n o g o l o n o m ( 1 . 1 1 3 ) k o 'rin is h g a e ga ed i. M a s h q sifa tida s h u m a s a l a n i n o g o i o n o m s h a r t g a m o s k e l u v c h i i k k i n c h i m e t o d ( 1 . 9 8 ) y o r d a m i d a y e c h a y li k . B u n i n g u c h u n (1 .1 1 3 ) shartni ( 1 . 9 5 ) va ( 1 . 9 6 ) k o ‘ri- nih sla rga keltirib olin a d i: - х + а в = ^ с кЯк = ° - ( 1 . 1 2 3 ) S h a r tn i n g so n i b itta lig id a n fo y d a l a n ib ctk —» ck ek a n li g ig a o ‘t.ildi. Y u q o r i - d agid an t o p a m iz k i q = - 1 , c 2 = a. Harakat te n g la m a la r in i ( 1 . 9 8 ) orqali y o z ib o l i n s a y a n a o ' s h a ( 1 . 1 1 9 ) fo r m u la la r g a k c li n a d i, fa qat o x ir g i t e n g l a m a sifatida а в = л t e n g l a m a n i qoMlash kerak. M a s a l a n i b i r i n c h i m c t o d b il a n y c c h i l g a n d a а в = x m u n o s a b a t n i n g o ;z i d a n g i n a f o y d a l a n i l g a n e d i , s h u s a b a b d a n ik k i n c h i m e t o d h a m h u d d i o ‘sh a y e c h i m n i beradi. 1-bobga mashq va savollar 1. E rk in lik d a r a j a l a s i n i m a ? 2. Q a n d a y s i s te m a in e r s ia l s istem a d e y i l a d i ? 3. B e r il g a n L a g r a n j f u n k s i y a l a r i u ch u n h a r a k a t t e n g l a m a l a r i n i to p in g : a) L = - i f - — c f \ b) L ~ - t i f \ c) L - - в " + - s i n 26 0 2 - c o s 0 . 2 2 2 2 2 4. Q u y id a g i L a g r a n j f u n k s i y a l a r i g a m o s k e lu v c h i h a r a k a t te n g l a m a l a r i n i toping: a) L = ^ - ( ф 2 + ^ в 2 + ф в ) + 31 cos(p\ b) L - tX'X ^ ; c ) L = - s ! \ - x 2 + A ( x ) k - ( p ( x ) ; d) L = x 1 -( 5. Q u y i d a g i L a g r a n j f u n k s i y a s i b il a n x a r a k t e r l a n u v c h i s i s t e m a l a r n i n g t e z l a n i s h l a r i n i toping: 34 a ) L = - \ j \ ~ ~ q 2 + q\ Ь) L - ^ i■' + 2 r 6 + ^ ; 1 ^ c) L = - ( x ' + \ ‘-) + ( x y - y x ) . 2 6. To 'liq h o si/a n i ta s h la b y u b o r i s h y o 'li b ila n e k v i v a l e n t L a g r a n j f u n k s i y a s i n i tuzing: a) L' = —(q + t ) 2 \ b) L ’ - ^ (q + q ) 2 ; c ) L' = x y - y x \ tl) L' = txx ; e) £.' = f) L' = — (2 a x t + a 2t~). 7. L ^ > L' - L + — f ( q . t ) a lm ashtirish n a ti ja s id a Eyler— L a g r a n j te n gla- dt m a l a r i n in g о ‘z g a r m a s l ig i n i к о ‘rsating. 8. G aliley a lm a s h t ir i s h la r i r —> r ‘ = r + \ t b a ja r il g a n id a erkin z a r ra n in g L a g ra n j f u n k s i y a s i e k v i v a l e n t k o 'r in i s h g a o 't is h in i к о ‘rsating. 9. D e k a r t k o o r d i n a t l a r i bilan q u y i d a g i c h a b o g 'l a n is h d a b o 'lg a n k o o r d i - n a tl a r d a m o d d i y n u q ta n i n g L a g r a n j f u n k s i y a s i n i tuzing: a) x = R(rp + siru p ), у = R( \ - c o s ( p ) ; b) х = (р + в , у = - : U c) x = J $ n c o s ( p , у = yf^rj sin (p. - = ^ ~ . 10. L = - y J x 2 L a g r a n j f u n k s i y a s i g a m o s k e l u v c h i t a ’s i r S = j L d t uchun x = c/chA + r s h A , f = f/shA + r c h A alm ash tirish b a ja r ilg a n d a S = ^ L (x )d t = j L ( q ) d r bo'lis hin i к о 'rsating ( A — о ‘zg armas, q = d q i d r ). 11. B ir j i n s l i og'irlik m ciydonid a j o y l a s h g a n g o r i z o n ta l ch iz iq d a h a r a k a t - la n u vch i i n i va ve r t ik a l ch iz iq d a h a r a k a t l a n u v c h i т э z a r ra la r n in g L agranj f u n k s i y a s i n i tuzing. U ia r n i n g o r a s i d a g i m a s o f a o ' z g a r m a s v a a g a te n g ( 1 . 3 - r a s m n i n g b ir in ch isig a q a r a n g ) . 35 / / / / / / / / / / / / / / a t m , rasm. 1 1 - , 12- va 1 3 - mas hqlarga oid. 12. 1 . 3 - r a s m n i n g o ' r t a s i d a k o ' r s a t i l g a n t e k i s m a y a t n i k u ch u n L a g r a n j f u n k s i y a s i n i tuzing. 13. 1 .3 -r a sm n in g o'n gida ko'r satilgan m a y a t n i k uchun L a g r a n j f u n k s i y a s i n i toping, m, n u q ta x o 'q i b o 'y i c h a ix tiyo r iy h a r a k a t q i l a d i . 14. M a s s a s i m va uzu nligi I bo'lgan m a y a t n i k n i n g osish n u q ta s i a ) v e r t ik a l te k islik d a a radiu sli a y l a n a b o ' y i c h a o ' z g a r m a s J b u r c h a k t e z li g i b i l a n h a r a k a t l a n m o q d a ; b ) у о ‘q i bo ‘y i c h a a c o s y t qon un bo ‘y i c h a h a r a k a t q ilm o q d a . S h u s i s t e m a l a r n i n g L a g r a n j f u n k s i y a l a r i n i toping. m x a У 1.4- rasm. 14- m ashqga oid. 15. 0 ‘z a r o ta ’s i r e n e r g i y a s i U = kv, - Г- bo ‘Igan i k k i z a r r a d a n i b o r a t m e x a n i k s i ste m a n in g L a g r a n j f u n k s i y a s i n i tuzing, h a r a k a t ten g l a m a l a r i h a m d a r , ( / ) , r , ( 7 ) la r n i toping. 2 -b o b . H A R A K A T I N T E G R A L L A R I H a rak at ja ra y o n id a sistem aga kirgan m o d d iy n u q talarn in g holati o 'z g arad i, sh u n g a k o ‘ra, ularning u m u m la s h g a n ko o rd in ata la ri qj va tezliklari q i h a m o ‘zgarib boradi. A m m o shu kattaliklardan tuzilgan va fizik jara y o n d a v o m id a o ‘z qiym atini o ‘zgartirm aydigan kattaliklar h a m m avjud, ular saqlanuvchan k a tta lik la r deyiladi. M a te m a tik t a ’rifdan boshlaylik. T a ’rif b o ‘yicha f { U q \ , q 2, - , q nA \ , 4 2 i - A * ) = C\ (2-1) funksiya dL d 3L . - = 0, d q t d t dcjj I,.... a - ( 2 . 2 ) differensial te n g la m a la r sistem asining birinchi integrali yoki h a ra k a t integrali deyiladi q ac h o n k i Ч\, Чг^-^Чп, Ч,^Ч\^Ч 2 ^-^Ч„ larning o ‘rniga (2.2) te n g lam alarn in g yech im in i q o ‘y g a n i m i z d a / f u n k s i y a m i z o 'z g a r m as so n g a a y lan s a . U n i n g son q i y m a t i m a s a la n in g b o s h l a n g ‘ich shartlariga b o g ‘liq b o i a d i . H a rak at integrallari saqlanuvchi kattalikning yan a bir b o sh q a nom lanishidir. (2.2) te n g la m a la rn in g b irinchi integral- larining soni bir n e c h ta b o i i s h i m im kin. U m u m iy hoida s erkinlik darajali sistema 2s — 1 ta harakat integraliga ega b o i a d i . Saqlanuvchan kattaliklar fizikada m arkaziy rollardan birini o ‘y n a y d i. S a q la n u v c h a n kattaliklarning h am m asi h a m teng m a ’noga ega emas. M asalan, bir nec h ta saqlanuvchan kattalikdan tuzilgan ixtiyoriy funksiya y a n a sa q lan u v c h an kattalik b o i a d i , a m m o unin g m ustaqil aham iyati katta b o i m a y d i . Biror bir kattalikning sistema u c h u n qiym ati shu sistem aga kirgan qism lar u c h u n q iym atlarning yig‘indisiga ten g b o i s a , bu kattalik add itiv kattalik deyiladi. S a qlanuvchi k attaliklar i c h id a additivlik xossasiga ega b o i g a n l a r i a yniqsa katta a h a m iy a tg a egadir. Biz shu b o i i m d a 37 k o 'r ib c h iq ad ig a n saq lan u v c h i kattaliklar bir t o m o n d a n f u n d a m e n t a l xarakterga ega — u la rn in g kelib chiqishi fazo va v a q tn in g f u n d a m e n t a l xossalariga b o g ‘liq — ikkinchi t o m o n d a n u lar additivlik xossasiga ega. S aqlan u v ch a n kattaliklar fizik ja ra y o n la r haqida m u h im m a ’lu m o t beradi va k o ‘p g in a ho llard a m a s a la n i t o ‘liq yechishga b i r d a n - b ir i m k o n iy a t beradi. 2 .1 . Energiyaning saqlanish qonuni V a q tn i n g b i r jin s lil i g in i n g n atijasi b o 'i g a n s a q l a n u v c h i k a tta lik k e ltirib c h iq a r a y lik . B u h o l d a L ag ran j fun k siy asi v a q tg a o s h k o r a b o g ‘liq b o ‘l m a y d i , y a ’n i, L ~ L ( q , q ) b o ‘ladi. V a q t g a bo g M iq lik L agranj funksiyasiga fa q a t k o o r d i n a t a la r q.(t) o rq a lig in a kiradi. S h u n i h is o b g a olib, L a g ra n j f u n k s i y a s i n i n g vaq t b o 'y i c h a t o ‘liq h o s ila s in i t o p a y lik : ± L = d t I . dL 4> + d dL dt dq X d . 7 t q > dL d q , d чг-' . dL (2.3) d t о ./=1 d(jj Bu fo r m u la n in g o ‘ng t o m o n i c h a p t o m o n ig a o 'tkazilsa V . 3 L 7=1 d q , ■ 0 fo r m u la hosil b o ‘ladi. Q avs ichidagi kattalik en ergiya deyiladi: dL • - L. (2.4) (2.5) U vaqt o ‘tishi bilan o ‘zg a rm ay d ig an , saqlanuvchi kattalik ek a n . O lin g an (2.4) m u n o s a b a t en ergiyan in g saqlanish qonuni deyiladi. E nergiya saqlanish q o n u n i fizikadagi eng m u h i m t u s h u n c h a l a r d a n biri ekanligini hisobga olib un i y a n a b ir y o ‘l bilan keltirib chiqaraylik. E yler - Lagranj te n g la m a la r i (2.2) ni q, ga k o ‘paytirib quyidagi h olga keltiriladi: 38 B u yerdan k o ‘rinib turibdiki, agar L vaqtga o sh k o ra bog'liq b o ‘i- m a sa L = L(q, q) energiya h arakat te n g lam alarin in g birinchi integrali, y a ’ni saqlanuvc han kattalik b o 'l a r ekan. E nergiyaning son qiym atiga k elganim izda u (2.5) ga kirgan tra y e k toriya va tezlikning boshlang'ich qiymatlari orqali aniqlanadi. Energiyasi saqlanuvchi sistem alar kon servativ sistem a la r deyiladi. U l a r qatoriga yopiq sistem alar kiradi. Agar sistem a vaqtga b o g ‘liq b o ‘lm ag an tashqi m a y d o n d a h arak at qilayotgan b o i s a bu h o id a h a m u n in g Lagranj funsiyasi vaqtga b o g ‘liq b o ‘!maydi, vaqt yana bir jinslidir, energiya saqlanadi. 2.1.1-misol. Energiy a harakat t e n g l a m a s i n in g birin ch i integrali ek a n lig in i isbot qiling. fo r m u la g a k ela m iz . Qavs ichidagi ifoda kin etik va p o te n s ia i en ergiy alarn in g y ig 'in d isi - t o ‘liq energiyadir. Isbot: mv = — d U ( r ) d r ( 2 .7 ) ( 2 . 8 ) (2 .9 ) ga ten g , o ' n g t o m o n d a g i IJ = t / ( r ( f ) ) fu n k s iy a u c h u n ( 2 . 10 ) Download 132.13 Kb. Do'stlaringiz bilan baham: 
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