《变分原理在加速结构中应用的论证》由会员分享，可在线阅读，更多相关《变分原理在加速结构中应用的论证（5页珍藏版）》请在金锄头文库上搜索。
1、 26 1 2002 1C o mme n t s o n Va r i a t i o n a l t e t h o d U s e di n Ac c e l e r a t i n g S t r u c t u r eW ANC Ia n- Fa( IE Mitutr of HiO EnerPhysics Chinese Academy d Sei-nem.BeUing 1 0 39 Chma)Ab , ct E e variauonal em for geneTal acceleraung g UUC M e m studied in detail . U e m ter wish
2、es topoh t out mm ermm m previous papem and explains ,ome importm t pomu m the app licaIion d the varia-tional e m aim .Ke y wo,variational method , acceleratine smlcttm , eleckomap etie fh ld1I n t r o d u cU o n le variaUonal method has Mgh curacy and needs small m E space .TEemfom , many peopleus
3、ed vam uonal method study accelerating sm scu m in the past decades l ,especially for disk-loadedsh mtlue in lim e . Although they gave good mSUIts, them were some enwm in these papem about the vm a- donal theoE 7 For example , Masao Nakamum 1 did not consider the non-metal condition at the two ends
4、 oftlm disk-loaded stmctm, and the is other mathematic em m in the pmeedm to pmve the vam uonal for-mula . Wans Boci l2j pointed out these pmblems and obtained the vmtationd formula wtth new method . Hearst asm m s that the elecmagnetic fleld satid es Mazwell efm Hom , metal boundary condition on me
5、taland periodic condition at the two ends of one cell , then the vad ational fOIT E mla SJ = O is pmved by usingthe metal boundaE y condition E x n = O and periodic condition . is not what we want to pmve . E econCt way is to pmve that the fK Ids wiH satisfV Mu well equatiom and boundaz7 condition i
6、f SJ = 0 . A nd the metal boundar7 condition E n = O cm t k used a known COZ E diHon because the trial funcdondoesit need satisfy this condition as in Ref . 3 . “is explained in the m xt section - Yao4 also de-E tved the vm ath nal formula . He Srst pmved that tbe Mu well equation will h sm d ed by
7、selecting O mspecial function ( Eq . ( 5 . 2 . 14 ) in Ref . 4 ) on metal bounda1 7 , “x n=,-x n= 0 ,( 1)hem n is the unit vector ouw aM nom al -SUE face d the accelerating struetum - 1Een he concluded thatthe metal boundaE 7 condition will k sa id ed tbr an arbitmzy by using Maxwell eq m Uon that i
8、s dem edfrom Eq . ( 1) . Iis is inconsistent M th Eq . ( 1) that mqui s a special function The vad ational method for geneml accelemting structure is studied in this paper and some imponantpoints on the application of the variational tbmulla wi ll h explai ned .2V ar i a t i o n a I T h eo f o r E l
9、ec t r o m a g n c F i et dMSt r u ct u r eTTm Maxwell equations for fleld with time dependence of ejM in vacuum space areV xZ o H= j kE ,( 2 )V x E = -j IzZ o H ,( 3) th boundaE 7 COIldiHons on metal mvxHxn= O or m xE = 0 ,( 4 )Reeeived 4 -m u7 M O 1 V01.26 , No.1Jan. ,2 2HIGH ENERGY PHYSICS AND NU
10、CI ZAR PHYSICS74- 78 1wbem k = i s t p mm P“ n m t Z o = F; h tint u m rW e caE n 1 obtai n the foll owi ng eq1 uE at “i o E n 1 for m ag neti c f E 1 k M e dld f hE m 3m t he abov e M axw ell eq uations ,V x( V x H )-t H =0 .( 5 ) e g eld Z o H, wh i c h sau des Eq .( 5 )an d th e bou nd - COE -H n
11、( 4 ) , cm be f ou n d by m ak ing tbe l -long var iati onal form E m mu mH )=j P m) mH ) .-t ( z mE ) .l - h - 0 w m E n E b el hw .U t mdef IE E e H ,H wh ic h 6 gfE es equationSJ ( Z o H )=0 ,( 7 )w bems mam su t iom Consi der H , i c h is n eu lyeq ud toH l bu t sli gh t d ev iated hm H Ibywb ei
12、s a azt E itm zyh n ction ,H=H I+a .( 8 )Sub-utuUng Eq.(8) in Eq.(6) , obtain1 + = j i m( H1 + “ ) )(J(HI + a,)Because H I S aHd Eq.(7) ,we have1 + a, ) | . . .s O( 10)Sta b-tituh a Eq .(9) imEq. ( 10) ,“bu inj p ! ( v h x , )From the vector ide t n t 6 i6e. V ( A x B )= B - V x A -A - V x B ,( 12)
13、we obtain the following equa iom by using ( V X H l ) (V x H J ) instead of A and , ( . ) instead d Bin Eq .( 12) , ( V x , ) ( V x H E ) = - V ( V x H J -x , + - V x (V x H I Y,( 13)( V x , ) . ( V x H 1) = - V ( V x E J X + . v x ( V H I ) .( 14)h tung F 4 .( 13. 14) in Eq . ( 10 . we obtainj V X
14、(V X H 1) - - vd v - j v H 1) , d V + CCz 0 ,( 15)whem CC. mem co tex enjte d ttz e rest pa . IEe seeond tem in the above equaIion can k rew it-ten asj v (V X H I ) d V = j (V X H I ) x mds ,( 16)wbma is the unit “ C outward norm I M the m da e S . meM ore Eq. ( 15) can h exF d aj V (V FEt ) -KV EI
15、- vd V JmH J x , mdS + CC= 0( UsiE tbe idem tiu“ A - B x C g C - A B = B C x A ,( 18)tbe second tem in Eq. ( 17) can fuE ther be trad om ed “j mH I ) x , mdS = j ( ,H I ) , - dS =j ( , ) H I ) dS( 112 e above inteF ion can be divided into hm in F neral . One ie on the metal bounda and another : - + , )(V x (H1+ a, ) - K Z(H1+ ,)(H1+ “ )|d V.( )76is on the non-metal boundaE Y Suppoee H l has the same tm nsvem compom nt H in Eq . ( 5
