Konar, S P

Sengupta, P K

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M la n J .  Phys. 64B (2), 138-143 (1990)Hamiltonians for magnetic fields in simple geometriesS P Konar and P K SenguptaDepartment of Physics, Burdwan University, Burdwan-713104,West Bengal, IndiaReceived 27 June 1989, accepted 2 1 July 1989H am ilto n ian  representations o f m agnetic  fie ld s  using m agnetic  coordinate systems  have recently been used by some authors in plasm a physics (Boozer 1982a , b, C arey and L ittle joh n  1 9 8 3 , Channel 1 9 8 4 ). M a g n e tic  coordinates are used in the  stu d y  o f m agnetic  surfaces (M iy a m o to  1 9 8 0 ) w h ich  have a relevance in to ro id a l fusion  d evices. A s a firs t step to w ard s  an actual s ta b ility  analysis, w e  w o h id  w r ite  th e  m ag n etic  fie ld  line eq uations in th e  H am ilto n ian  canonical form . Tpis  requires th a t w e  fin d  a H a m ilto n ia n  fo r th e  system  in term s o f su itab ly  defined  'canonical coordinates and m o m en ta '. Once th is  is done, th e  K A M  theorem  or the C h iriko v  crite ria  can be applied  to  study th e  s ta b ility  o f th e  system .W e  consider tw o  re la tive ly  sim ple c a s e s : (1 )  a h e lica l m agnetic  fie ld , (2 )  th e  fie ld  due to  a long s tra ig h t s teady current.A  d irec ted  line elem ent in orthogonal cu rv ilin ear coordinates u^, u„, u„ is/s ^—- c I h -^f~ Cj|h 2^u 2 "1' II *Aand th e  m agnetic  fie ld  e.,B„, are u n it vectors in the d irectionso f increasing u^.The fie ld  lin e  eq uatio n  isr f s x B -0  (1)This  leads toh idu  1 _h jjdu jj __h A ^B , B ~ “  B ~W e  d efine  th e  'canonical coord inates  and m om enta' asq-q(U i, Uj) p =  p (U i, u * )and th e  'canonical t im e ' r = U g  1 3 8(2)(3)Hamlltonlons for magnetic fields In simple geometries 139R equirem ent o f Local sing le valuedness of H  together w ith  the condition  { T . i f = 0  leads to  the fo llo w in g  possible sets o f q, p and H, these sets being  connected to  each other by canonical transform ations (Jan ak i and Ghosh 1987)j ) — j  hjhjBgdUjj,  f f i  hghgf l iduji i ) ^ e ~ ^ a r p a - ~  ^ h iha Bj dUj ,  ^ hj^hyBgdUj^“ «  J*ai i i )  p » ~ ^ i t  q#" j  ihB®8*^^B< a ~  j“1 “1iv ) p« - U a ,  q * - - ^  h ihgBgdUi, H * -  -  ^ h^h.,B..du^(4 )(5 )(6) (7 )The H a m ilto n ia n  is such th a t the equations of the fie ld  lines are consistent w ith  the canonical equations8H dp . S H ^ dqSq dr ' Sp dr(8)W e  use c y lin d rica l p o la r coordinates r, z, 0 w ith  e , xe^  - e , etc ; th is  convention  is som ew hat d ifferen t from  th a t usually  used. But th is has a defin ite  advantage  in th e  case of to ro id a l geom etry  (W o o d s  1 9 8 7 ). HereU i, Ujj, z, 0 ; h i  — 1, fig -- l.h g  r and B^ =  Br,B b - B * , B „ - B , .  (9 )The fie ld  com ponents fo r a helical m agnetic fie ld  are (B le w e tt and Chasm an 1977)B r - 2 B „ [ l . . ( k r ) - / i ( k r ) ]  sin (d ■ kz)B *-=  - 2 B o / i ( k r )  cos (0 kz)B . = ^  / i ( k r )  cos ( d - k z )* kr(1 0 a )(10b)(1 0 c )w here B,, is the transverse o n -a x is  fie ld  am plitudes and L ,  U  are zero-th  order and 1st order Bessel functions o f im aginary argum ents. In the case o f a helical m agnetic  f ie ld  w e  have the firs t choiceq ,^ r ,  P i-^B .dz , H ,^\rBrdz ( 11)In  v ie w  o f eqn. (1 0 ) , th is  leads toU (k r )  sin  ( 0 - k z )H i - 2 B „ [ k r  lo (k r )-~ lj^ (k r )] cos ( f l - k z )  From  th e  f ie ld  line  eqs. (2 )  it fo llo w s  th a tkz)140 S P Konar and P K Sengupta^ : Z®’- = kr'’ -  -1 ® '"i-de Bg 111 k r jc o s  (e - k z )^  rB_gdo” B / k r “(1 2 a )(1 2 b )(1 3 a )(1 3 b )%■l i ( k r )  cos ( O - k z )  | l  - k ^ ) (1^ )^\For a h e lix  o f constant p itch  (k - c o n s ta n t ,  as w e  have assum ed above) w e  can  take 0 - 0  fo r z ^ O  w h ic h  m akes (o -  kz) 0  fo r a ll z. In such a case eq. (1 4 )  reduces to/ i ( k r ) [ 1  } k ’^ r^]A lso  under the sam e boundary  cond itio n ,^^^~-J^i.^2B„[kl ..(kr)-fkM„(kr)-kli(kr)]  o q i orU sin g  th e  recursion re la tio n s  (W a ts o n  1 9 3 8 )l’o ( x ) - / x ( x )  < - , . ( x ) - l , . ( x )x l« (x ) +  n l„ (x ) - x l„ .  i ( x )  w e  can w r ite- 2 B „ [ k / , ( k r )  +  k “f / i ( k r )  -  { k r / , ( k r ) -  l i ( k r ) ! ]  (1 I k>*r*‘ ) l i ( k r )(1 5 )(1 6 )From  eq . (1 5 )  and (1 7 )  w e  getSH,SrIrr-(1 7 )(1 8 )Hamiltonians for magnetic fields in simple geometries 141N o w , in  th e  H a m ilto n ia n  th eo ry , p and q are regarded as independent variab les , b oth  being  ( im p lic it )  fu n ctio n s o f th e 't im e '.  Therefore in dea ling  w ith  the other can on ica l eg u a tio n , w e  have to  regard p^ and r as independent.From  eq . (1 2 a )  and  (1 2 b ) w e  then g e t* i (k '') ]  sin  ( e - k z )& pi * i ( k r )  ■ c 6 s ( f l - k z )_ r B ,_ d r _  .B f  de(1 9 )W e  note  th a t th is  eq . (1 9 )  ho lds  independently  o f th e  boundary cond ition  (0 kz) =  0 . W h en  th is  bou n d ary  c o n d itio n  is taken  in to  consideration , eq. (1 9 )  is tr iv ia lly  8Hsa tis fied  because both  and are then zero.6piLinear c u r r e n t  :W ith  o ur cho ice o fr, U j . - z  and u^ — 0o ._p a _  . gB- e„ (20)(21)0—tH i-^rB ^dz[S in ce  B , = 0 ,  the integral should be a function  o n ly  o f r and 0]dt do 277 1 r do r -  do JSince fo r th e  f ie ld  linesdo do6pj S r S rdo 6HI f  th e  can on ica l e q u a tio n  ~  ~~s}^w e  should  have ^  =  0  so th a t f i  is a function  on ly  o f 0. SrW e  m ay take w h ere  <  is a constant.SH SfThen — ' ^ = / i -  =  0 ,  because p i  is function  o f z  and r w h ile  f i  is a fu n ctio n”Plo n ly  o f 0 and ^ = 4 ^ = 0  for the fie ld  lines. S o  th e  other canonical equation ,de don am ely , =  is also tr iv ia lly  s a tis fied .do SpjBy a canonical transform ation , w e  m ay o b ta in  <1 = 11, = z ;  |> - p ,=  -^ h ,li,B ,iiu ,In  r 4  constant 277H^= -^hihgBjdUi=  ” \  rB *dr =  f , ( z ,  0) ,  because B * - 0142 S P Konor and P K SenguptoT h is  set o f q u an tities  also leads to  th e  canonical equationsSH,anddp2. do ■dqa.doSp“ , both  sides being zeroSp both sides being a g a in  zero.W e  can o b ta in  tw o  m ore a lte rn a tive  form s by s im ila r canonical transform ation s.W e  have sh ow n  th a t ( i )  in the case o f a he lica l m agnetic  fie ld  (g iven  by eq. 1 0 .) , th e  canonical co o rd in a te  q and m om entum  p and the H a m ilto n ia n  H  g iven  by eqs. (1 1 )  and (1 2 )  and  ( i i )  in th e  case o f th e  m agnetic  fie ld  due to  a long  steady current, q, p and H  g iven  by eq. (2 1 )  lead v ia  the fie ld  line equatio n s  to  the  correct canonical e q u a tio n . W e  m ay, therefore, conclude th a t th e  canonical equations them selves represent th e  fie ld  lin e  equatio n s  in th e  respective cases.Acknowledgm entsT he  authors sincerely th a n k  P rof B D asgupta  o f th e  S ah a  In s titu te  o f N uclear Physics, C a lc u tta  and  P rof S  K C hakravarty  o f th e  D epartm ent o f M ath em atics , B u rd w an  U n ivers ity  fo r he lp fu l d iscussions. O ne o f th e  au thors  (S P K ) is gratefu l to th e  U n ivers ity  o f B u rd w an  fo r fin an c ia l assistance.Hflmlltoniflns for magnetic fields in simple geometries 143ReferencesBoozer A H 1982a Phys. Fluids 25 620Boozer A H 1982b Phys. Fluids 26 1 288Blewett J P and Chasman R 1977 J. Appl. Phys. 48 2692Carey J R and Littlejohn R C 1983 Ann, Phys. 151 (1)Channel P J 1984 Los Alamos Nat. Lab. Pep. No. LA-10048-MS Chirikov B V 1979 Phys. Reports 52 263Goldstein 1980 Classical Mechanics 2nd edn. (New  York ; Addison-Wesley)Janaki M S and Ghosh Gautam 198 7 ] . Phys. A20 3676Miyamoto K 1980 Plasma Physics for Nuclear Fusion (Cambridge, Ml A : MIT Press)Watson C N 1958 A Treatise on the theory of Bessel functions 2nd. edn. (Cam bridge: Cambridge University Press) p 79Woods L C 1987 Principles of Mognetoplosma Dynamics (Oxford Clarendon Press) p 223

Hamiltonians for Magnetic Fields in Simple Geometries

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Hamiltonians for Magnetic Fields in Simple Geometries

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