Applications of Lagrange Equations

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1 Applcaton of agang Euaton Ca Stuy : Elctc Ccut ng th agang uaton of oton, vlop th athatcal ol fo th ccut hown n Fgu.Sulat th ult by SIMI. Th ccuty paat a: 0.0 H, 0.00 H, 0.00 H, C 0.0 F, C 0. F, 0 Ω, Ω an a 00 n (00 t). a (t) C C

2 Ca Stuy : Svochan ng th agang uaton of oton fo th ctly vn vo-yt. Con a vochan actuat by a oto wth two npnntly xct tato an oto. oa wth png T ω, T Data: ng th block aga of pannt-agnt ynchonou oto, a llutat n th Fgu abov, vlop th SIMI aga. Sulat th vo-yt wth th followng paat: 0. Ω, 0.00 H, M H, a -, an J kg-. 00 n (00 t) 0 n (00 t).

3 Ca Stuy : A Th-ha annt Magnt Synchonou Moto Apply agang uaton of oton to tuy th ynac of th followng panntagnt ynchonou oto. Sulat th yt. T oa c b ω, T c b oto Stato a a Soluton: a b c ω a b c T a b c Th ultng agang uaton a:

4 Th total kntc ngy nclu kntc ng of lctcal an chancal yt n patcula ( ) cc n ) n( n J cb bc bb ca ac ba ab aa Th lf- an utual nuctanc a fn by th ubcpt, an th flux tablh by th pannt agnt not by Ψ. cc co co co n n n J cb ac ca ac cb bc bb ba ab ca ac ba ab aa Snc th no png n th chancal yt, th potntal ngy 0 Th pat ngy houl b foun a a u of th hat ngy pat by th lctcal yt an th hat ngy pat by th chancal yt ( ) W obtan Th agang uaton, whch a xp n t of ach npnnt coonat, la to fou ffntal uaton

5 aa b ( ) ( ) a b ( ) ( ) ab a ( ) ( ) ac J a ba ca a ab ba bb bc co b cb ac bc b ca cb cc co c c c c ω co co a a ω co( ) ω co( ) b c T b c ot: a b b ( t) ( t) ( t) M M M co co( ) co( ) Data: ng th block aga of pannt-agnt ynchonou oto, a llutat th Fgu abov, vlop th SIMI aga. Sulat a th-pha, two pol pannt agnt ynchonou oto wth th followng paat: 0. Ω, 0.00 H, l 0.00 H, H, a - (--A-), a -, an J kg-. fo th tannt analy by upplyng a balanc th-pha voltag t M 0. ot: Th utual nuctanc btwn nuoally tbut tato wnng ab, ac, ba, bc, an cb a poc functon of an hav th avag valu (DC coponnt). Aung th agntc fl unfo, an akng u of th fact that th agntc ax a plac by (/), on conclu that th DC coponnt of ab, ac, ba, bc, an cb. ab ac ba bc cb co ( ) Sybol: Slf-nuctanc of th tato wnng l Stato lakag nuctanc l oto lakag nuctanc cou fcton coffcnt J Euvalnt ont of nta Magntc of th flux lnkag tablh by th pannt-agnt.

6 Ca Stuy : A Two-ha Inucton Moto Fn th athatcal ol ung th agang uaton of oton fo a two-pha nucton oto oto. oa T - ω, T a a b b - Couplng - a a b b a W ay wt th followng uaton: b a a b a b b, - T 6

7 7 Th xpon fo th total kntc, potntal, an pat ng a gvn by: ( ) 0 n n co J bb bb ab ba ba ab aa aa co n n co

8 [( ) n ( ) co ] J n co co co co In t of ognal valu, w hav a a co b co 0 n co n co ( ) ( n ) ( n ) ( n ) ( co ) ( n ) J a ( n ) ( co ) a a a b a b a co b n [( aa bb ) n ( ab ba ) co ] Tl b a Data: ng th block aga a llutat n th Fgu abov, vlop th SIMI aga. Sulat th vo-yt wth th followng paat: 0. Ω, 0.00 H, H, a -, an J kg-. a 00 n (00 t) b 0 n (00 t), a 00 n (00 t) b 0 n (00 t). a a a a a b 8

9 Ca Stuy : A Two-ha Inucton Gnato Fn th athatcal ol ung th agang uaton of oton fo a two-pha nucton gnato. oa wth png ω, T T a b a b a b a a b a b b, T p W ay wt th followng uaton: 9

10 0 Th xpon fo th total kntc, potntal, an pat ng a gvn by: ( ) 0 n n co J

11 [ ] 0 co n n co co n co co n co co co n co J In t of ognal valu, w hav [ ] p a b b a b b a a b b b b a a a a a a a a b a b a a a a a T J co n co n n co n n n co

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