UBC Theses and Dissertations

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UBC Theses and Dissertations

Periodic orbits of the second genus for the crossed orbit problem of the helium atom Milley, Hermon Reginald 1941

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L£  [fur A % . n  PERIODIC ORBITS OJ? THE SECOKD GEMJS FOR THE GROSSED ORBIT PROBLEM 01 THE HELIUM ATOM  Herman. R e g i n a l d M i l l e y  A The3is Submitted i n P a r t i a l F u l f ilment of t h e R e q u i r e m e n t s f o r t h e D e g r e e of M&STER OF ARTS i n t h e D e p a r t m e n t of MATHEjIAT IC S  THE UNIVERSITY OF B R I T I S H COLUMBIA APRIL, 1941  CONTESTS  I . PRELIMIMARY: THE FIRST GENUS ORBITS. § 1. I n t r o d u c t i o n  Page ( 1 )  § 2. The D i f f e r e n t i a l E q u a t i o n s  "  (2)  § 3. The E q u a t i o n s o f D i s p l a c e m e n t  "  (4)  I I . THE SECOND GENUS ORBITS. § 4. D e f i n i t i o n o f Second Genus O r b i t s  Page ( 6 )  § 5. The D i f f e r e n t i a l E q u a t i o n s  "  (7)  "  (8)  § 6. The E q u a t i o n s o f V a r i a t i o n and  their Solutions  § 7. N o t a t i o n g 8.. The P e r i o d  o f t h e Second Genus O r b i t s  I 9. The S o l u t i o n s References  !l  (15)  11  (17) «  (17)  »  (28)  -I-  PBRIODIC ORBITS: OF THE •" i  FOR  SBOOHD GENUS  THE  GROSSED ORBIT PROBLEM' OF THE HELIUM ATOM  I.  PRELIMINARY.  §1. I n t r o d u c t ion« It for  i s proposed h e r e t o construct  the s e c o n d genus o r b i t s  a s p e c i a l c a s e o f the problem, d i s c u s s e d by Dr.  Buchanan  1  i n h i s p a p e r ) "Crossed O r b i t s i n the R e s t r i c t e d Problem, of T h r e e Bodies w i t h R e p u l s i v e and A t t r a c t i v e . F o r c e s . " The d e a l t w i t h i s that designated i n the l a t t e r  case  as Part I I ,  Case I . The  p r o b l e m considered d e a l s w i t h the m o t i o n of  two  i n f i n i t e s i m a l b o d i e s which a r e a t t r a c t e d by a f i n i t e b o d y but r e p e l l e d by each, other, the nature of the f o r c e s b e i n g Newtonian ( i . e . , , obeying: the inverse s i m p l i c i t y , the two  involved  s q u a r e l a w ) . For  i n f i n i t e s i m a l bodies w i l l be c a l l e d  " e l e c t r o n s " and the f i n i t e body the " n u c l e u s " . A p a r t i c u l a r s o l u t i o n of the problem i s that i n which the  e l e c t r o n s r e v o l v e i n c i r c l e s w i t h the n u c l e u s a s centre  and  r e m a i n d i a m e t r i c a l l y o p p o s i t e . Two  tjrpes of o r b i t a r e  obtained when the e l e c t r o n s are d i s p l a c e d from t h e i r c i r c u l a r motion. In part I the e l e c t r o n s and  equidistant  remain d i a m e t r i c a l l y opposite  from the nucleus. In part I I the distances  the e l e c t r o n s from the nucleus are equal, t h e i r d i f f e r by 180°,  but  of  longitudes  t h e i r l a t i t u d e s are the same. The  p a r t i c u l a r case which i s common to parts  I and I I —  that i n  which the l a t i t u d e s are zero — it  I s c o n s i d e r e d i n p a r i I I and  i s d e s i g n a t e d a s c a s e I.. I t i s i n t h e v i c i n i t y of these,  o r b i t s t h a t we s h a l l make o u r c o n s t r u c t i o n o f t h e s e c o n d genus orbits,. I n s e c t i o n s 2 and 3 a b r i e f o u t l i n e of t h e r e s u l t s obtained  i n " G r o s s e d O r b i t a . . . . " w i l l be made, s h o w i n g t h e  method used i n c o n s t r u c t i n g the. f i r s t  genus o r b i t s u p o n w h i c h  t h e work of t h i s t h e s i s i s to- h e b a s e d §2* The D i f f e r e n t i a l E q u a t i o n s . . Taking a rectangular n u c l e u s and d e s i g n a t i n g x  l»  s y s t e m of a x e s w i t h o r i g i n a t t h e  the. c o o r d i n a t e s o f t h e e l e c t r o n s * b y z  Yl» z-i  w  2:2* J2* 2>  e  h a v e f o r t h e f o r c e f u n c t i o n . "U,  of t h e s y s t e m :  •awhere k^ = r a t i o of r e p u l s i o n to. a t t r a c t i o n , , r  *  ±  r  2 -  r  s i  2  2  ( x x  L  ( 2 | ]  x  2  + y  2  + ?2 x  4 2^2it-  L  l - 2 )  2  2  *  z  2  2 )*  i Cji-y2)  2  4  2  C^l-22) ]*  and. t h e u n i t s o f s-paae and tIme have b e e n s o c h o s e n t h a t t h e g r a v i t a t i o n a l constant equals unity* The e q u a t i o n s o f t h e m o t i o n a r e t h u s  F r o m t h e s e e q u a t i o n s and a c o n s i d e r a t i o n of the. c o n s t r a i n t s o n t h e m o t i o n of t h e e l e c t r o n s a s o u t l i n e d i n t h e  introduction,  we  o b t a i n t h e two  Part l i t  y  2  " ~  z  2  = ~  S£  = —  i^,  y  - -  y»  z  As we we  2  y  z  E  l'  r  l~ 2  r  =  ^ 12'  l*  x  z  2  ~ i -  a r e n o t c o n c e r n e d w i t h t h e d e v e l o p m e n t of p a r t  s h a l l c o n s i d e r i t no  ( 3 ) , we  need c o n s i d e r t h e m o t i o n  e l e c t r o n o n l y . On s u b s t i t u t i n g t h e s e r e l a t i o n s  equations  (2)  and  E  in  t r a n s f o r m i n g the r e s u l t i n g equations  t h e v i s - v i v a i n t e g r a l t o c y l i n d r i c a l c o o r d i n a t e s by t h e  and sub-  stitutions:: we  obtain;  ,*  A  A- 0 " -t- 3~A-'& ' — 3."  =  —  we  get:  a  l  n  /• h^B~  t h i . last relation  g  /i," •=  g2  ~  —  These e q u a t i o n s  where  '  O  t h e i n t e g r a l of (5 b) b e i n g U  ^  =• / -  I  further.  By v i r t u e of r e l a t i o n s of one  d i v i s i o n s of t h e p r o b l e m , v i z ;  A  & •  to a l i e n a t e  f l  o.  __  have t h e p a r t i c u l a r  ^  ff'  a </*-*-< I  solution  (5).  ~4~ §3. The E q u a t i o n s o f D i s p l a c e m e n t  and t h e i r  Solutions.  B y means o f t h e s u b s t i t u t i o n s  we o b t a i n f r o m e q u a t i o n s ( 6 ) t h e " e q u a t i o n s o f d i s p l a c e m e n t " , that is,. the equations g i v i n g the displacements from the plane o r b i t g i v e n by s o l u t i o n s  'f t-o*he i + p < W  ( ? ) . They a r e :  -  1  f % ^ r ' j - 1  f  Yf V  f V 5  - ^ c i * i j [ e t»rrj - s , Y ^ ' T " - i r ; *  The s o l u t i o n s  ••• J •• - J  of t h e s e e q u a t i o n s w i l l g i v e t h e p e r i o d i c  o r b i t s of t h e f i r s t genus of w h i c h t h e r e a r e t h r e e t y p e s , e a c h t y p e b e i n g c h a r a c t e r i z e d b y i t s p e r i o d . The p e r i o d s a r e determined  o  f  (  9  ,  :  .  by a c o n s i d e r a t i o n of t h e e q u a t i o n s of v a r i a t i o n  h f - °  ;  -°  w h i c h have t h e t h r e e s e t s of g e n e r a t i n g s o l u t i o n s , v i z : Case I :  f> - f\Onit Period:  Case I I :  ^  x  1  -+  f^A^ ^  ^ jr  ] ) o d ^ i  Period:  -f- £  ^  ^  J^L i r r a t i o n a l *  Case I I I :  p  e  r  i  o  d  : T  =  M ^ ) = ^ ( - f  -5Gorresponding found  t o case  t o e x i s t o n l y when  I , a solution  of e q u a t i o n s  (9) i s  ^ ~ £> . T h i s s o l u t i o n f o r m s t h e  " g e n e r a t i n g s o l u t i o n " f o r t h e s e c o n d genus o r b i t s t o be c o n structed. The  first  genus o r b i t s f o r c a s e I a r e o b t a i n e d b y s e t t i n g -  +  where t h e PV  f, £ 1-  are variables  On s u b s t i t u t i n g of  £^ on each s i d e  i n equations  f* a r e c o n s t a n t s . s  ( 9 ) and e q u a t i n g  coefficients  of t h e r e s u l t i n g e q u a t i o n , a s e r i e s of  d i f f e r e n t i a l equations  w i l l a r i s e , which,  and t h e  of t h e form  when i n t e g r a t e d  sequentially,,  determine  the v a r i o u s The  P• a n d <T„- . V * initial conditions f'(o)  = /  ;  f>(o) ~- o  serve t o evaluate the constants a r i s i n g from the i n t e g r a t i o n s , The  results are: ^ Ccru Z  -o  4- £ ( %  C  "~ i Caj-^T  (10)  -6II-  THE  SECOffl) GEMJS ORBITS 2  34. D e f i n i t i o n of Second Genus O r b i t s . ^ Suppose we  i n which the t  have a s e t of d i f f e r e n t i a l  equations  a r e a n a l y t i c i n the. a r g u m e n t s , do not c o n t a i n  e x p l i c i t l y , and  a r e p e r i o d i c w i t h p e r i o d T. The  i n g e n e r a l , a f u n c t i o n of t h e p a r a m e t e r admit the p e r i o d i c  £  period i s ,  , I f these  equations  solutions  \ - 9-  t)  (€;  h a v i n g the. p e r i o d T, t h e n s u c h s o l u t i o n s a r e s a i d t o be of first  genus. Mow  where  let  £.'  0  i s c o n s i d e r e d a s a f i x e d c o n s t a n t and  v a r i a b l e parameter. above d i f f e r e n t i a l  the p e r i o d i c ^  A  as  a  When t h e s e s u b s t i t u t i o n s a r e made i n t h e e q u a t i o n s we  t h e r e a r e no t e r m s i n d e p e n d e n t admits  o b t a i n a s e t i n y^ i n w h i c h of y^ o r  i \ . If this  set  solutions =  ; -t)  having the p e r i o d  H b e i n g an i n t e g e r , t h e n t h e %.  = 0-  (8  O J  a r e s a i d t o be. of t h e second with  the  "A  , the second  solutions -t) +  (€ ^ A j ±) a  genus. S i n c e t h e  genus o r b i t s a p p r o a c h  vanish t h o s e of  the  „7f i r s t genus as  X  approaches z e r o .  §5. The D i f f e r e n t i a l In  equations  Equations.  (9) make t h e s u b s t i t u t i o n s  £ ^ C„ (' + X j  where £ o c c u r s  where t h e z e r o s u b s c r i p t i s a t t a c h e d t o f, to  £•  merely  i n d i c a t e t h a t t h e y a r e t h e o r i g i n a l v a l u e s of t h e s e  tities for  j f , £,  explicitly,  as g i v e n b y e q u a t i o n  quan-  ( 1 0 ) . ( M o t e : t h e r e a r e no s o l u t i o n s  s e c o n d genus o r b i t s i n t h e p l a n e . ) We t h u s  obtain the equations:  3  x  +-fi 1 *J<+v-'  f . C o n A W o V ^  3  j  j  -8t h e terras I n d e p e n d e n t o f p and q h a v i n g d r o p p e d o u t b y r e a s o n of e q u a t i o n s (9)„ §6. The E q u a t i o n s  o f V a r i a t i o n and T h e i r S o l u t i o n s .  The. e q u a t i o n s  of v a r i a t i o n  equating t o zero t h e l e f t  /  o f (11) a r e o b t a i n e d b y  side:  r ( ^rj^i-(^3'=m^ /tt  Since these  (  •• - J  r3c  two e q u a t i o n s  a r e independent,  their  s o l u t i o n s w i l l be c o n s i d e r e d s e p a r a t e l y . Consider differential  f i r s t , equation  (12,a). B y the t h e o r y of l i n e a r  equations w i t h p e r i o d i c c o e f f i c i e n t s ,  get p a r t i c u l a r  s o l u t i o n s ' t o t h i s equation by d i f f e r e n t i a t i n g  p a r t i a l l y i t s generating solution arbitrary  % e may  (10,a) w i t h r e s p e c t t o t h e  constants which are contained  i nthe l a t t e r but  w h i c h do n o t a p p e a r i n t h e o r i g i n a l d i f f e r e n t i a l e q u a t i o n . I n t h i s c a s e , two s u c h a r b i t r a r i e s The two p a r t i c u l a r  u  How,  then  0  and £ .  solutions are  o  since  2  occur: t  [ £ e )=  .  1(^J . ^  j  ^f\^ )]^(^^)  *  0  o  Also,  j  -(12^  + ^rci-f-^^)£ 2  ^  1  2  )  -9We  therefore  obtain  Sir] - Cc<r»*f**-+J • [~ ^ This being a p a r t i c u l a r stant factor,  since  c o n s t a n t l a t e r , and  solution,  i^) w i l l be  we  r  " may  *<<^ d i s r e g a r d the  multiplied  by  an  our  particular The  solution  other p a r t i c u l a r  a periodic  function  For,  £  (in  since Z  by  plus  corresponding to solution  of  ^  (12,a) w i l l  ~C t i m e s a n o t h e r p e r i o d i c  enters into ), we  undetermined  take  1r as  con-  /°  b o t h e x p l i c i t l y and  i . consist  of  function. implicitly  have  2 where t h e  brackets enclosing  ferentiation On solution  denote e x p l i c i t  dif-  only.  p e r f o r m i n g the  indicated  differentiations,  this  becomes:  1- A t  ytr) here  ^ £  ^Lt)  Is g i v e n  by  (13.3 and  X  appears i n the  2£  £  differentiation  3  y  )  -10-. I n o r d e r t h a t (13.1) and (13.2) may c o n s t i t u t e a m e n t a l s e t -of s o l u t i o n s order that the general  Kj_ and  ~ • of e q u a t i o n  funda-  (12 , a) , t h a t i s , i n  s o l u t i o n of (12,a) s h a l l he  being a r b i t r a r y constants, the fundamental  deter-  m i n a n t of t h e s e t must n o t be z e r o . This determinant i s :  ^  CrJ  ^Cr) + /\<p(V f- A C -4>Lx) 5)  How  i t has b e e n shown  t h a t t h e v a l u e of t h e  determinant  of s u c h a f u n d a m e n t a l s e t i s c o n s t a n t . We c a n t h e r e f o r e compute t h e v a l u e of D most c o n v e n i e n t l y b y s e t t i n g to  f  equal  zero. Thus  ± - f •»--- -  O  +-  1  • ,  I t t h e r e f o r e f o l l o w s t h a t t h e most g e n e r a l s o l u t i o n of e q u a t i o n (12,a) i s  and  Lp  being  , of c o u r s e ,  periodic  ( ~x.Tr)  The above method c a n n o t be used i n t h e s o l u t i o n of equation  ( I 2 , b ) b e c a u s e o f t h e a b s e n c e of a g e n e r a t i n g  solution,  -11or r e t h e r , b e c a u s e i t s g e n e r a t i n g s o l u t i o n i s i d e n t i c a l l y 61 z e r o . T h i s • e q u a t i o n , a f o r m of H i l l ' s s o l v e d by making the standard  equation,, ' w i l l  be  transformation (14.1)  i n which  /  /• p  , V - «  ^  *  . *•  v 4 >  "• a.  The v a r i o u s  and  /Z£  will  be d e t e r m i n e d so as t o  fulfill  t h e f o l l o w i n g two c o n d i t i o n s : ( i ) The p e r i o d i c i t y c o n d i t i o n  (14.2) ( i i ) The  initial-value condition:  (o)-=t j n^r°) = ^ The s u b s t i t u t i o n of (14.1) i n e q u a t i o n  ^ (12,b)  i  —  yields:  or  sAs b e f o r e , we  e q u a t e t o z e r o t h e c o e f f i c i e n t s of  £  E a c h s u c h e q u a t i o n w i l l have f o r i t s c o m p l e m e n t a r y f u n c t i o n t h e s o l u t i o n of "J Coefficient  of  ^ £  :  f IT  .J  ^- ^* . •__2_ p ^' ^  _ „  -o  .  -12whence  a.  A p p l y i n g c o n d i t i o n s ( 1 4 , 2 ) , we g e t  /  bmce G  2  .—  , i ngeneral,  0  i s n o t r a t i o n a l , we must have  = 0. Theref ore  nr  -  0  Coefficient  i  of £  or •t • • lip*.  f  ifii / '  ^  t  r  0  y  How i n o r d e r t o i n t r o d u c e no t e r m s p r o p o r t i o n a l t o ' Into our s o l u t i o n ,  i t i s evident  that the constant  c  t e r m on  t h e r i g h t s i d e of t h i s e q u a t i o n must v a n i s h . That i s , . r  A Integration  °  gives i «  ,  .  ._  _!_ g «1  and a p p l y i n g i n i t i a l a n d p e r i o d i c c o n d i t i o n s we o b t a i n ±IT -  whence, Therefore  C-? = 0,. and <-> '  0  o  2  =  • 4 —  -13C o e f f i e l e n t o f fi*..  + > ^ ( l ^  Substitution f o r  ,^  To e n s u r e t h a t v cause t h e constant equation  ^  2  c  *J c ^ ^ J -  j^/ir  -  a  , /zr y i e l d s  s h a l l have no n o n - p e r i o d i c  t e r m s on t h e r i g h t  terras,  we  s i d e of t h i s l a s t  t o v a n i s h . That i s , —  /S  -  — 1 - -  whence  ^  1  »  ^  3 J^- ( >  ;  Then  3(^+3)(^-«J  cr  S(tJ^.-3)(^-H]  M  x  + UL - J*e~ ^ /(fO+fc) /(>/* t  As b e f o r e ,  i t c a n be shown t h a t  0^=0,  and t h a t  has t h e v a l u e 5  33/* 3  2  - 264/U j 264/u. + 288 8/^(l—/<)(4—/^)  2  I n s i m i l a r f a s h i o n we c a n f i n d as many more  ji-  S  and  fifj  -14as w i l l  d e t e r m i n e f3  and  if t o any d e s i r e d d e g r e e of a p p r o x -  imation.  Having  now  o b t a i n e d the: p a r t i c u l a r  the other p a r t i c u l a r - L  i n it and  &  s o l u t i o n q of  (14.1),  s o l u t i o n i s o b t a i n e d by r e p l a c i n g i  , t h a t i s , the conjugate  by  of q i s a l s o a  7) particular  solution  T h u s , t h e g e n e r a l s o l u t i o n of e q u a t i o n _ (fj  Tr  .  (12,b) i s :AZ  (  s,-' /7,, -'  'J  where  and  2  v^ ) If  d i f f e r s from  v ^" ^ o n l y i n t h e s i g n of  t h e s i g n s of b o t h  'C  and  <^ ..  ^ a r e changed i n ( 1 4 . 4 ) ,  q does not c h a n g e . T h e r e f o r e , b e c a u s e of t h e p a r i t y of cos and  s i n Z,  i t must be e v i d e n t t h a t t h e c o e f f i c i e n t s of  c o s i n e terms i n  a r e a l w a y s r e a l and  terms always p u r e l y  imaginary.  The  fundamental determinant  w i l l be computed f o r l a t e r u s e .  those  the  of t h e s i n e  of t h i s s e t of s o l u t i o n s  I t i s -.  'C  -15-  l  4T  /A  being constant In value, i t s evaluation w i l l  e f f e c t e d by s e t t i n g 1  How and  v  %  v ^ ) (0)  (D(o) = -  by e q u a t i o n  =  0. 2  v ^ ) ( 0 ) = 1,-  =  v^(o)  be most e a s i l y -  =  ^  •t  by e q u a t i o n (14.2) , i i , ( C +  t  , V  ^  ••  • j  (14.4).  Hence a power s e r i e s  T h i s completes  in  £  (14  t h e s o l u t i o n s of t h e e q u a t i o n s  of  v a r i a t i o n ( 1 2 ) . These s o l u t i o n s are t h e complementary f u n c t i o n s of e q u a t i o n s determined  (11) whose p a r t i c u l a r  i n t e g r a l s are next to  be  as power s e r i e s i n °\ .  §7. d o t a t i o n As  t h e a l g e b r a i c e x p r e s s i o n s become too" u n w i e l d y  so o b s c u r e  t h e methods of a t t a i n i n g c e r t a i n r e s u l t s  s u b s e q u e n t c o n s t r u c t i o n s , we letter"  n o t a t i o n of D r .  The  i n the  "foundation-  Buchanan  n o t a t i o n c o n s i s t s of s y m b o l s of t h e  g( > K ) and  s h a l l employ t h e  and  )(  S ^'  form.  J  r e p r e s e n t s power s e r i e s i n )\ h a v i n g f o r t h e i r  sums of c o s i n e s or s i n e s r e s p e c t i v e l y . Of t h e two  coefficients parentheses  -16I n t h e s u p e r s c r i p t s , t h e f i r s t has two e n t r i e s : a n i n t e g e r 0, 1» 2,  ...,  f o l l o w e d by t h e l e t t e r  e  d e s i g n a t e s b o t h t h e l o w e s t power of  £  parity i n £,  or  arguments  while the l e t t e r  of t h e c o s i n e s  o f "5 r e s p e c t i v e l y . The  e  or  a. The  integer  i n t h e s e r i e s and t h e o  denotes t h a t the  o r s i n e s a r e even or odd  second p a r e n t h e s i s  multiples  contains  an  integer  w h i c h d e n o t e s t h e amount by w h i c h t h e h i g h e s t m u l t i p l e of i n t h e arguments coefficient  of t h e t r i g o n o m e t r i c t e r m s o c c u r r i n g i n t h e  of any power of  £• e x c e e d s t h a t power.  I n t h e f o l l o w i n g work we obtained  by d e l e t i n g t h e l e t t e r s  parenthesis cosines  s h a l l use t h e m o d i f i e d e  and  o  s i n c e i n bur c a s e t h e arguments  i n the  notation first  of t h e s i n e s  and  a r e n e i t h e r e x c l u s i v e l y even n o r e x c l u s i v e l y odd  m u l t i p l e s of T> * We  may  have o c c a s i o n  given foundation  letter,  a l s o to a d j o i n a s u b s c r i p t to a  i n w h i c h c a s e we  are c o n s i d e r i n g a p a r t i c u l a r An  understand that  s e r i e s of t h a t  we  type.  Example. • ' .  "d  0 ) ( o )  =  e (  c  ^  c  ^  "  «3K-—j %~~~\  F o r f u t u r e r e f e r e n c e , we may  ^  C  /V/  ^ &  c  ^ 1<  K  '  . c ^ ^ r 1C  *" " " ' "  ^  note that i n t h i s  notation,  -17§8. The P e r i o d  of The O r b i t s of t h e Second Genus.  F o r t h e f i r s t genus o r b i t s , t h e p e r i o d I s air i i -  n  x>  3  c  which i s  Tjr- —  K}+b) ~- T 2  The p e r i o d  In  t.  of t h e s e c o n d genus o r b i t s i s t o be,by  definition, N' T IT b e i n g Therefore in  ( I + a power s e r i e s i n A J  5  i n t e g r a l . How t h e p e r i o d i n order  t h a t our f i n a l  of t h e s o l u t i o n q i s s o l u t i o n s be p e r i o d i c {  IS , we must h a r e t h e p e r i o d ^  ~  ^ * -  • T,  l  f  r  a power s e r i e s i n ^ J  as r e q u i r e d by d e f i n i t i o n . A l l v a l u e s does n o t h o l d a r e  in Z  of £ f o r which t h i s  excluded.  §9. The S o l u t i o n s o f E q u a t i o n ( 1 1 ) . A f t e r s u b s t i t u t i n g f o r the value i n equation  (ll)  r  o f f> ( e q u a t i o n  we. a r r i v e a t t h e f o l l o w i n g :  V  (10)).  -18where  'oo +  f -  £> (i \. ^  In order t o integrate in  A  ^ )  ^ ^  equations  , we p u t  j + - - -  ( 1 5 ) a s a power  >  On e q u a t i n g c o e f f i c i e n t s of l i k e powers of }\ r e s u l t i n g e q u a t i o n we o b t a i n a s e r i e s in  p j and  solutions that  i n the  of d i f f e r e n t i a l  equations  q.. h a v i n g f o r t h e i r c o m p l e m e n t a r y f u n c t i o n s (13.4) and ( 1 4 . 4 ) r e s p e c t i v e l y .  determined that  p j and  p r o c e e d t h e n as  q. w i l l have t h e p e r i o d indicated.  the  I t w i 1 1 be shown  t h e ~Y- and t h e c o n s t a n t s of i n t e g r a t i o n w i l l 4  We  series  be so T2  -19-  Knowing t h e complementary f u n c t i o n f o r the f i r s t  of  t h e s e e q u a t i o n s , we may s o l v e i t c o m p l e t e l y by t h e method of variation  of p a r a m e t e r s .  Thus,  where t h e a r b i t r a r i e s have been a s s i g n e d t h e s u p e r s c r i p t s  1  fi) m  o r d e r t o a s s o c i a t e them w i t h  and k^ 1 *  will  be a s s o c i a t e d  p^ and  q^. I n g e n e r a l  w i t h t h e s o l u t i o n s of  J  k| '  p. and  q  Here D i s the fundamental determinant the  of § 6 and has  value D = c o n s t a n t = 1 4 a power s e r i e s i n £ . Employing the f o u n d a t i o n - l e t t e r n o t a t i o n , these  may be w r i t t e n :  equations  -20  3> /,„  tJ"->  whence, upon I n t e g r a t i o n ,  3 - k!" -  the  4 ^^y,(^^r'j] J  d's r e p r e s e n t i n g power s e r i e s i n £ w i t h c o n s t a n t  coef-  ficients. S u b s t i t u t i o n of these function  expressions  i n the complementary  yields  ZD  +• A  On r e d u c i n g ,  ft^sTj  *-ffi<??  and n o t i n g t h a t t h e t e r m s i n At  The c o m p l e t e s o l u t i o n a t t h i s  stage w i l l  cancel o f f ,  now be  -21-  In order that  be p e r i o d i c , t e r m s c o n t a i n i n g  T  as a f a c t o r  must be e l i m i n a t e d . T h a t i s ,  Therefore the p e r i o d i c s o l u t i o n s f o r  p^ and  q-j_ a r e  fi y The use  a r b i t r a r y constants  of t h e I n i t i a l  KV  1  may  be e v a l u a t e d by  the  conditions  Applying these c o n d i t i o n s  t o t h e above e q u a t i o n s  I t i s found  that 1  K* * The  s  ' 0;  =  -  p e r i o d i c s o l u t i o n s t h u s become  ' 1-  (20) 3  "  I n e q u a t i o n (18) we 1  Kg ^  and  ^  the process determine  . Another  such r e l a t i o n s h i p w i l l  of I n t e g r a t i n g  t h e s e two  have a l i n e a r r e l a t i o n  p^ and  q  2  connecting  be f o u n d  which w i l l  in  uniquely  c o n s t a n t s and hence f u r t h e r s i m p l i f y ( 2 0 ) .  -22Coefficient  of A  We o b t a i n t h e s e t :  i n which  ~?  f". otSj [&  +f, E  £  *  ^  Upon v a r y i n g the parameters, there r e s u l t  D and A  b e i n g t h e d e t e r m i n a n t s g i v e n i n §6.  C o n s i d e r t h e l a s t two o f t h e s e e q u a t i o n s . We I n v e s t i g a t e t h o s e p a r t s of y,  2  o j ) which contain  , w i t h the view t o o b t a i n i n g a supplementary  shall 1  Kg ^  relation  to ( 1 8 ) .  The t e r m s of  and  (2) Q, i n w h i c h we a r e i n t e r e s t e d a r e t h o s e  -23containing  £  and  , and a r e shown i n t h e f o l l o w i n g  table  Coefficient  of£.('+*)  Coefficient  Our two e q u a t i o n s may now be  of fiO + S) f ' ^ ^  written  Aki* 2  Q,[ ^  and  2  Q^ ^  i n t h e above  b e i n g t h o s e p a r t s of  not  included  table.  Removal o f c o n s t a n t t e r m s g i v e s  or O (23)  -24since  ^ rjlb  0,  o t h e r w i s e we w o u l d h a v e t h e c a s e f o r  a n a l y t i c a l c o n t i n u a t i o n of t h e f i r s t genus o r b i t s . E v i d e n t l y we must e x c l u d e a l l v a l u e s of  O  equations We  f o r which the  determinant  7-  (')  a of  £  (24)  <  (18) and  (23) i s e q u a l t o z e r o .  have, therefore,.  K  1  d  2 ^  1  ~ 3 ^  which gives f o r equations  a n d  %  ~  d  1  4 ^  (20),  (2) (2) Q  When t h e s e v a l u e s have b e e n s u b s t i t u t e d , there  The  t h e c o e f f i c i e n t of  We of  v  A  of  here  . The  s i g n i f i c a n c e of t h i s f a c t  i n the l a t e r  s h a l l now  equations  P  ' and  result:  coefficient  be a p p a r e n t  in  I s t h e same as t h a t of 1^ i n will  development.  complete  t h e i n t e g r a t i o n s of t h e l a s t  (22) . They a r e of t h e  form:  pair  -25e  On  integrating,  -kr^^rvr'tv-^v.'sr''' Substituting  these r e s u l t s  we a r r i v e a t t h e p a r t i c u l a r  i n t h e complementary  ^ vys-"l  •  e  yield,  pair  q  gives  2  i s :  ^) e' M~ ^• f  r  <26  r  of e q u a t i o n s ( 2 2 ) . They  on s o l u t i o n , a r e l a t i o n i n v o l v i n g  Integration  function  integral  Thus t h e c o m p l e t e s o l u t i o n f o r  C o n s i d e r now t h e f i r s t  (25)  V _ We have,  will  -26The p a r t i c u l a r  I n t e g r a l i s , therefore,.  •f-  or  and we g e t f o r t h e g e n e r a l s o l u t i o n : :  (27)  The c o n d i t i o n f o r p e r i o d i c i t y w i l l a A second  -t-  K  + ^«  give the r e l a t i o n =  °  (28)  s i m i l a r r e l a t i o n which, w i t h (28), w i l l unique(2) E " and / , w i l l a r i s e i n t h e next i n t e g r a t i o n 2  ly  determine  in  t h e same manner as d i d e q u a t i o n ' ( 2 5 ) . M o r e o v e r , t h e c o e f -  ficients and  ~Y  f  0  of  and  ~Y w i l l he t h e same as t h o s e of  ^  i n ( 2 3 ) . T h e r e f o r e the f u n c t i o n a l determinant of these  two e q u a t i o n s w i l l  he t h e d e t e r m i n a n t ( 2 4 ) .  Thus we h a v e K  2  2  )  =  a  n  d  d  ^  = 4  2  )  '  C o m p l e t i n g t h e i n t e g r a t i o n s i n (27),. we f i n a l l y at  the general solutions f o r p  2  and  ~h  q  c  2  ;  arrive  -27-  On I m p o s i n g I n i t i a l c o n d i t i o n s i n e q u a t i o n s i n t e g r a t i o n c o n s t a n t s may Proceeding determining .... K^ 0  those  and  ~Y-  evaluated.  s i m i l a r l y , we  the succeeding  can c o n t i n u e the process  p.  J  and .  q. J  , t h e two  of  unknowns  b e i n g d e t e r m i n e d by txva r e l a t i o n s s i m i l a r t o  of e q u a t i o n s  determinant  be  ( 2 9 ) , the  will  be  (18) and  (23).. M o r e o v e r , t h e  (24).  H  functional  -28-  REFEREHCES 1) - D. Buchanan:  "Crossed O r b i t s  i nthe Restricted  problem, o f  T h r e e B o d i e s w i t h R e p u l s i v e and A t t r a c t i v e F o r c e s . " —Rendiconti  d e l C i r c o l o Matematico d i Palermo,  Tomo LV, ( 1 9 3 1 ) . 2K  D- Buchanan:  "Periodic Orbits  Straight-Line Equilibrium  o f t h e Second Points  Genus near t h e  i n t h e Problem of  Three B o d i e s . " — P r o c e e d i n g s of t h e R o y a l S o c i e t y , 3) . H. P o i n c a r e :  A, V o l . 114, (1927)  "Les Methodea U o u v e l l e s de l a M e c a n i q u e  Celeste."  —Tome I , C h a p t e r I V .  D. Buchanan:  "Asymptotic S a t e l l i t e s  Line Equilibrium Points —American Journal  near t h e S t r a i g h t -  i n the. P r o b l e m o f T h r e e B o d i e s "  o f M a t h e m a t i c s , V o l . 4.1, ( 1 9 1 9 ) .  4) . F.R.Moult on: " D i f f e r e n t i a l E q u a t i o n s " — "Periodic  Orbits"  5) . F . R . M o u l t o n :  "Periodic  O r b i t s " - C h a p t e r I , § 18.  6) . F . R . M o u l t o n :  "Periodic  O r b i t s " — Chapter I I I , Part I I .  7)  "Periodic  Orbits  F.R.Moulton:  8) . See 2) above.  ; l  —  Chapter IV.  Chapter I .  — Chapter I I I , § 51.  

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