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The limiting case of periodic orbits near the lagrangian equilateral triangle solutions of the restricted… Hamilton, Rognvald Thore 1939

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C A S E 0 ? P E R T D D I 0 OBDISS S H E LAGRINGIAN EQUILATERAL S R I A N G L E SHE  NEAR  LIMITING  OF SHE R E O S R I G S E D T H E S E B O D I  PROBLEM.  SOLUTIONS  L £  THE L I M I T I N G C A S E NEAR THE  OF P S R r O D t C  COLUMN?  THREE BGD3T P R O B L E M .  fair  Rogtirs.lel  There  Hamilton  - s C o -  A ffe.esir. c u b m i t t o d f o r t h e D e g r e e o f  in  t h e Department  of  MATHEMATICS  The U n i v e r s i t y  of B r i t i s h  April,  1939.  ft' 7  ORBITS  L A G M r f O I Ap EQUILATERAL T R I A N G L E  OP THE B E O T R I 0 2 E D  5  Columbia  "IRE LinXSIfTG HEAR THE LAOEA'TCIMT  OF PERIODIC O R B I T S EOtfTLASERAL  TRIAKGLE  SOLUTIONS  OF THE R E S T R I C T E D THREE BODY PROBLEM.  ~oOo~  Pago E n u n c i a t i o n o f t h e Fx-oblem General Conditions f o rt h e Existence Periodic Orbits Three DiKiorsrinnal ^ r b i t s Existence of  of  o f P e r i o d /SIT  that P l a n e O r b i t s o f P e r i o d S i r  Construction o f the Plane  14 1?  Orbits  ~oCo~  DIAGRAHS  Tfiroo  Dimensional  Piano  Orbits  Orbit  f o l l o w i n g pace IS f o l l o w i n g page- S3  THE HEAR THE  L i : " T J I I G 3AGE f ? P E R I O D I C  CEBITS  LAGEAIJGIAfl E Q U I L A T E R A L T B I ANGLE DGLUTIGMG  01? THE  RES-SRI02EB T H R E E BODY E H O E L E K .  T h c p r o b l e m c f f i n d i n g t h e motions, o f t h r e e m a s s e s u n d e r 'the  influence of t h e i r  solvod  cm7.y f o r s p e o i a 1  The  circlett o the  and other  forever  the  There are  relatively  that the triangles  smr^ec .  certain  placed  r e l a t i v e t o t h e two  by  of three b o d i e s refors  t h i r d mass i s i n f i n i t e s i m a l w i t h  tvo.  s o l u t i o n s o f the first  at rest.  restricted  L a g r a n g e i n . 1772.  the  i n the  thst for  o f them w i t h  masses, i t v i l l i  The.five  such points  problem w h i c h were  vertices  of  zero remain  are  discovered  equilateral  p l a n e o f motion o f t h e  >  respect  2sro o f t h e s e s o l u t i o n s d e m a n d  Denote t h e r e m a s r e s  I t v a c texmd  to  points such that i f •  at one  finite  t h r e e bodies l i e a t the lying  been  o f t h e m^sres r o t a t e about each o t h e r i n  i n f i n i t e s i t i a l h o d ? be  velocity  a t t r a c t i o n s has  cor-diticms.  r e s t r i c t e d problem  c a s e srhere two  the  gravitational  by  1 - jj,  a n d >  ,,0335.. t h e s o l u t i o n s  finite  vrhere jn ^ • were  unstable,  a n d f o r ^ t ^ .0985.. t h e y r e r e  In t h e l a t t e r displaced a l i t t l e and  case-, i f t h e i n f i n i t e s i a a i b o d y i s from t h e e x a c t  given a s n a i l Telocity,  this point.  v e i n t of- a  i t - i l l  She d e t e r m i n a t i o n  near the Lagrannian  stable.  solution  oscillate  of periodic  around  oscillations  equilateral triangle points  f i r s t , made b y Bnolc i n 1 9 1 0 .  He f o u n d  v a r i o u s plane and  three dimensional o r b i t e , t h e p o s s i b i l i t i e s determined by t h e trains chosen foryU  was  being  , wherey.t < . 0 5 8 5 . .  b u t t h e l i m i t i n g case.' f o r ytf =s . C 3 3 5 . . , t h o u g h i t was Icnovrn t o b e s t a b l e . , ir'ap not, p r o p e r l y c o n s i d e r e d . I t i s the purpose periodic hero  by  orbits  f o r the H a l t i n g case.  f o rthe plane  Bncft.  o f t h i s paper t o c o n s t r u c t t h e S h e methods  orbi.tr a r e d i f f e r e n t  from those  They w e r e o u t l i n e d b y Dean D. B u c h a n a n ,  v/hece piidanc© . t h i s p a p e r w s s p r e p a r e d , nrm  t;inh t o express  and t o  used used  under  TJhom  1  tay t h a n k s .  .the. D i f f e r e n t i a ? E q u a t i o n s  •7c c o n o i d e r  in circles.  distance,  two f i n i t e  bodies  Shcir distance  rotating  about each  other  apart i s taken as the u n i t  and t h e u n i t o f time i s c h o s e n  t o make t h e  gravitational proportionality  the u n i t of  Further* of  equal,to u n i t y .  i s chosen so t h a t t h e manses  SI2.ES*  t h e two bodies--can be r e p r e s e n t e d She  motion of an i n f i n i t e s i m a l  rotating  the of  factor  axes, f  rectangular  c e n t r e of mass,•the the finite  the  bodies,  a n d { / i ^ i )  1 body  is referred to  •> t h o o r i g i n  ,  ^-plane  being  at  being t h e plane o f notion  and t h e r a t e o f r o t a t i o n s u c h  that  l a t t e r r e m a i n on t h e - g - a x i s . '"hen t h e e q u a t i o n s of. j a o t i o n o f t h e i n f i n i t e s i m a l body  are  -given  <££ _ ? 4rz  :  =  *U •  d>  where  U = ^(f "-*- YX)  Pertain  listing  She  2  . 2.41 -  <££ * ^ 2 -  +•  •SolfltjLons  above d i f f e r e n t i a l  •  (  1  )  e q u a t i o n s admit t h e i n t e g r a l  first  f o u n d by J a c c b i ; v i s . :  =  mum Vdt/  *dt'  Also,  zu  +  constant".  \dti  the equilateral triangle solutions  of Lagrange  a r e bnovrc t o b e :  f .  TT.  In  * -^  orbits  can  be obtained  near  the s o l u t i o n s  origin  ^  J  £ -  Q  the p o i n t readily  obtained  I . as those  on  for the point I I .  b y ©hanging t h e s i ? x i o f  /3  i n  near I .  i s t r a n s f e r r e d to the Lagrangian  t r i a n z1 e p oi n t  I'he  7 = -  c o n s t r u c t i n g c u r r:olut£@ns, we n e e d c o n s i d e r  the  The  ;  I . b y t he  equilateral  t r a n s io r a a t i on:  d i f f e r e n t i a l , e q a a t i o f t e become:  e expanrinnc  o f t h e r i g h t members, i n d i c a t e d , a b o v e  I n terms o f E a y l e r ' s  arc  ii  X -=-L f 2 1 ( 1 X,  ^5^[-~ ^ 37  2 u )  series,  (  —  a;  +•  Y  x  I  3/&  (.1  —  find:  2yt*}  y  - 3 3 ( 1 - 2 ^ ) y* +  %•  ~ 7 5 / 3 ( 1 - 2 > )% f  + ISSxy*- +  z  - 6 0 / 3 ( 1 ~ 2> )ys*  I, «  f o r \?b.ick'we  1 2 ( 1 ~ 2y*)  45/3(1 -  2/4  - 13 X**]  £ [ 3 / 3 ( 1 - 2,u ) % t ©y]  « -L^-S/Sx  1  ~ 6 8 ( 1 - 3 ^ ) x y - O/Sy*- +  * -1 [ - 2 5 / 3 ( 1 - 2 / * )s* +  12/3s*J  1 2 3 x y + 1 3 5 / 5 ( 1 - 2y« ) x y V S y 2  « I S S y s * - 6 0 / 3 ( 1 - 2/* ) x s * J  S —  S  —7.j  Z « z  f ^3-/3y «s 4- 3 ( 1 —  d = ^[-S3y z + 1 2 / z  agree w i t h  those  given  Buck.  fo  The  2)xs;J  Sx*?, - 3 0 / 3 ( 1 ~ 2 ^ ) x y s ]  T h e s e v a l u e r , do n o t e n t i r e l y by  ) /  region  )  of convergence i s e v i d e n t l y U n i t e d  singularities  o f -L a n d J L .  by the  3  if =  v/hero  o «  Hence  +  J-=  But  &f +  ( z +  y  f1 +  ( 1 •+-  q)"  1  2  4  (y +  s" +  <Jf +  2  x + /Sy.  1  « y*-  =  1 4-  n c  -»-. . .  .R.Cll. ._-». 1 ) . . • Q L . T T . . S I  -P  •ITil  T o X'f.nd- t h e c o n d i t i o n  n + that  1  putting  the limiting r, = z  we  test:  ~ 1 .< q < 1 -  i c By  f o r convergence, apply t h e r a t i o  obtain  tlie r e g i o n  1+  r <. SB (  the region  ,,  as  .  of convergence  f 4. ^4  i n the equation  q  o f convergence, f o r r  Similarly,  q  values of  f o r r^ i s  7/2.  (S)  We  no*? i n t r o d u c e  differential  :: = -r'£  where  £  will  the parameters  S  and  55 = ss £  ;  £,.  In the  equations •put t  ;  y = y'£  ;  4  serve as a s c a l e f a c t o r ;  t - t « 0  S  /2(1 + S ) T  i s a parameter  depending later  on  € :.  a n i 'the / 2 is i n t r o d u c e d  vrcrl-:. "lie  differential  e q u a t i o n s bocojae:  - 2/8(1 + $ ) D y **  ifz  D y + 3/3(1 r  -+  5 )Dz «  D a  2(1  (r  2(1+ S f  •+  & t  +  (D  2  t h e equations  - |)s-  [2/SD  - 3^(1  She indieial  [s/3D _  2 y  +  5^(1  «)] + S  Equations,  2^£ + JS^e* +-  2 ^  )]y =  (D* - | ) y »  and  we p u t  of v a r i a t i o n .  -  )  D s  S = €, -  0  Ihey are:  C 0  f i r t t fere m u s t b e s o l v e d s i a u l t a n e o u s l y . equation i s  )  l  i n "wMeh t h e a c c e n t s h a v e b e e n d r o p p e d ,  we o b t a i n  )  z  (  in the differential  X e+ l e V  (!, -»« Y € + I,£ -t-  = 2 ( l + 6 f (2 +  2  If,  t o ^4 rr.pl i f y  "heir  2ti"t i s : ' "ho  D  4  +  2.D" +  27/i (1 - ^  2  r c c t s of t h i s equation D  2  =  =fc / l - 3 7 / /  -1  ) =  0  arc given (1  -yU  by  )  i n B u c k ' s c o n s t r u c t i o n s , t h e r o e i r w a r e assumed t o dirtinct.  2he  p r o b l e m w i t h w h i c h v*e u r e  c o n s t r u c t t h e o r b i t s ^hen  be  concerned i s to  the r o o t s are equal;  that i s  trhen 1 - 27/f  (1  -  U  or  )=  = When t h i e ? a l u e  variation (D  2  fcr^u  ~  +  vritb. t h e V u l u o o f equations  s - i / ^  =  *  C 2 3 S  —  i n need, the e q u a t i o n s  of  beeone:  - f >r - (2/2D  (2/3D  0  + ( D  X  -  ^ ) y «  0  |)y =  €  choren, the general  of v a r i a t i o n  are:  solutions  of  the  2 as a c i n r + b C.OC-T + T ( C G I H T + - d C C S T )  y = a * s i n T + b c o s T + f ( c * c i n r -t- d ' c o c T ) 1  where  £'=  a , b , c , d, p, q , a r e a r b i t r a r y a n d  -^(/SSa*  b' =  d* =  4/2h) + ^  (7/2c + / 2 3 d )  ( V S a - /23b) + J L (-yS3c + 7/2 d )  ± ( 4 / 2 c ~ Vg3d) H e n c e we o b t a i n  the generating  I-  it = a s i n T + b C O S T  II,  : r = C ; y = 0 ;  ;  solutions:  y = a*sinT  -t-  b'COST  ;  a == 0  s -= p r i n / 2 T + q e o c / S T .  S h e c c s o l t s t i o n s w o r e o b t a i n e d bp a s s i g n i n g e p e c i a l v a l u e s to  thearbitraries.  $4  s  solutions  0  t o c-tioar t l i a t b y  talcing  a n d by c h a n g i n g t h e a r b i t r a r i e n , v;e c a n o b t a i n  having  and r e d u c i n g  We now p r o c e e d  t h e oaiac- p e r i o d a s t h e g e n e r a t i n g  t o t h e n when' <5 = £ =  C.  solutions  We may  x(o)  tahe as general  =  initial  conditions  y(c)  that:  s(o) «^s-  As wo c s e f r o m t h e e q u a t i o n , page 4 , t h e r e component  of force  therefore  t h e i n f i n i t e s i m a l body w i l l  com© t i a i e .  She  d i r e c t e d towards  H e n c e , we may  i sa  t h e xy-plane;  pass  through  i ta t  take  differential'equations  Cp-?) c a n b e I n t e g r a t e d a s  power- s e r i e s i n t h e p a r a m e t e r s <* , « » <v , <v , <* , 5 , et  5c a s s i s t i n t h e i n t e g r a t i o n i t t o s u f f i c i e n t  y *  +  s =  s„ +  y s w  +- y £. +  o, 5 +  '-.ho  0(  to put  .......  S £ 4- .......  will  enter  when t h e i n i t i a l  conditions are  11 The  initial  coefficients;  All  that  the other We  coefficients will  t. H  the  for  y, s ,  -  <f)*.+  T h e n , vrhen we differential  0  ;  f )y,  (D  2  - |)y-  put t h e s e r i e s  equations  and  take  for  x , y,  j ^ -  0  ;  (D  2  ~  2)2,.  •  C.  =  a s i n i +  b  y„ =  a'siirTH-  h ' c o s T + T ( c ' c i n T <+• d ' e o c t )  s„ -  p ssin/2T>  C O S T  +  T ( C  q c c e / 3 ~i  E i n T +  a,  i n the  the terms without  The s o l u t i o n s ' a r e :  ^  i n the  the c o e f f i c i e n t s of  get:  we  T = 0.  <§ , £ .  - (2/2D +  x  (2*H>  first  functions:  f.D -.  a  b e s e r e ^rhen  e q u a t i o n s and equate  powers of  Define  be a b s o r b e d by t h e  is-  p o t the above s e r i e s  differential equal  conditions will  d  cost)  S  t  g  s  in  v r h i c h 'the ~ r i a o i  on  -?.age . n  —  z  constants  d e p e n d ' on t h e o t h e r s  Open a p p l y i n g  the i n i t i a l  I  3  2-  4  4  as  conditions,  find:  4  P — ^ j  T-*~ II  ff6  q = 0  /a  Farther certain  be p o s t p o n e d  s i m p l i f i c a t i o n s c a n be made.  however, the  integration -will  that  factor  the  r i c h t  z, s o t h a t  t i l l  Oheortre a t t h i s  point  member o f t h e a d e q u a t i o n  (;•>.?)  i t s solution  will  have t h e f a c t o r  SMJ^ElMiy^x^sagtioaa^ ^ ?.h© g e n e r a l c o n d i t i o n s period-  2  1  that  the o r b i t shall  have a  are:  y(0) =  0  Dy(2)  -  r,(n) =  0  I> ( ) - Ds(C)=  PP= now  Khc« t h a t  s  o n e 'of t h e s e  2  -  Dy(C)«  y(S) -  C  C  i s a consequence o f the  a  has  ( p . 3}  o t h o r o by v i r t u o o f J a c o b l ' o I n t e g r a l "ho  corresponding  integral  f o r tho oouoiioro  of  race 7 i o  I? =  (Dsf +  (Dyf+  ( D s f - (1-t-Sf  +  Put  i n thic  Or- =  -v t i  •yhere  t^CO)*  e (  ) -J-  equation  C  =  Sy*~ 4  2 3  )  constant -  t h e imlttoo:  Dy = e ^ 4 - t i  ?  8/2Sit? +  ( i s 1, 3,  D a = ^  4  6  +  u  fe  _„. ,-0)  I h e n we c a n w r i t e  Hhir  agnation i o s a t i s f i e d  when  solve  for  aa a power g or l o o  Hence, i f came v a l i n e vjhon ry-p-rorf- t h e l a s t  « T=  for  r>,  5  0 .  C.  We  Also,  can,  i n . ti , t i , a  ore periodic 2  ::• =  so also w i l l  of thsperiodicity  2  and return u .  therefore,  , « , u , S $ £. 4  f  t o - the  We c a n , t a u c ,  conditions.  THREE lie  ORBITS  mmnsimKL  now  consider  the  corroo.pendin.E- t o She  period  Ar  possibility  the is  2  •without  loss  generality,  (p„0).  A.lt;o, s i n c e x,  v  4  inntead  of  With  the  obtain:  C =  h a  h b  +  C =  h  0 =  h a  0  +  ) 5  4- h^b  whore  p  of  b,  h^b  +  ( i«  my  d,  (p.lC)  are  retain  conditions? we  t ^ d + £p  ()  fiyc-  ".-=!:= C =-  the  d  4- I ^ d 4  the  (p.O).  the  a,  £  P | X  ?  4 i  may  put  scale  linear  apply  +  we  f  for -  £r ,+ £ 4  ••/,. . , 5;  d, S ,  c,  The o s  further,  orbit  factor,  functions b,  y,  c,  Ds,  <u  of  d,  By .  the. p e r i o d i c i t y  .....  +  (  solution  an  •  2  h a  a,  +  h 0  a 4- h pb 4- .....  3)  s  (2  conotrticting  because of  these- a l t e r a t i o n s , to  2)  we  initial-  equations  ((  conditions  a , fc, c>  (p.IS),  «* ,  HI T  =  initial  «*, r*\ -  of  /2TT  oocond generating  regards of  PERIOD  OF  3>  functions  £•  c-not ions  = S" =  are  e -  determinant  are  satisfied  by  r  in  h..  io different  fron  zero.  IB  Hence t h e f i r s ' t  four  e ^ u a t i c r t n a y be u s e d - t o e c l v e f o r  b , c , &, a r y<c*or e e r i e r i n 5 ,  a,  obtained  €-  nay then be s u b s t i t u t e d i n t h e f i f t h  obtain a function of & , £, alone. of  $  o~  i nthis  of  with  a , b. c, d, ^ ,  L  and converging  The can  .three  construct  exactly .  we c a n s o l v e  orbit  S  / - « ,  1  t h e same m a n n e r a s Bucfc h a s d o n e . rra f i n d  £  +•  49  i*7/? — c 0 * 2 / 2 T — *i r ' / ' T 11(49) 3(49) ~  234With  vanishing  e x i s t s a n d vre  '  £  s  for  small.  therefore  » - ^ i * ! cosS/ST+ ^ ^ s m g y s T  +J^L§ 12-  +  4-  the exception  t h e exnrotsrinn  for  to  e n a b l e s ue t o e z p r e n c  l&l s u f f i c i e n t l y  " 0  3C49>  in  for  equation  i t by a e s u t r . i n j j  In particular,  x  This  £o  the coefficient  an p o ^ e r s e r i e s i n e  dimensional  C M ' ; in  Since  function i snot zero,  a s a roarer s e r i e s i n £ .  all  The v a l u e r  of the coefficient  of  sinS/lT  z, e a c h o f t h e a b o v e t e r m s c a r b<  found  in  frois Back's c o r r o s ^ o n d i r . s . ones by  them t h e v a l u e s  The . s o l u t i o n  (p.4) the  above  velocity  will  i n t h e neighbourhood o f t h e p o i n t  c a n b e o b t a i n e d b y c h a n g i n g the. s i g n  /3  of  IX.  in  solution.  She " o r a c i o t c r  an  substituting  parallel  £. d e n o t e s t h e coupon e a t to the  iT-axis.  c o n v e r g e f o r \e{  sufficiently  neglecting  in  terms  L  lies  within  She above  the region  initio.!  solutions  snail.  o f t h i r d and h i g h e r  i l l u s t r a t i v e o r b i t was p l o t t e d  Shis o r b i t  o f the  f o r the- v a l u o o f convergence  powers  £ — k> (p.6).  I f -as n e e t h e p e r i o d .condition?  x =  to  3W+  vrhere  , v , % , have t h e f e r a s five  implicit  o f t h e n e we  can obtain  the  provided  already  fount! ( p . 1 1 )  functions o f the.a r b i t r s r i e s .  oech a r b i t r a r y t h a t t r i l l  conditions,  c o n v e r g e n t power s e r i e s satisfy  the  By in  periodicity  each o f t h e f u n c t i o n s : v a n i s h  with  arbitreries. p a r t i c u l a r , the a p p l i c a t i o n © fthe condition  In c ( P T T  )  C = s£t  -  #  vanish  obtain that  5t(c) »  sin2/2*rr  If to  Zo  io  s  noann  o\  J S +• - - . . -  ire o b t a i n  for  to  x 6>-v x £ + x 6 % - . • « "o  •J -  S i r i n a.--prlying t h e p e r i o d i c i t y  gives  -+ ©*,(  .....)  0 and i c d i v i d e d o u t , t h e e x p r e s s i o n  \-rhcn t h e p a r a m e t e r s v a n i s h .  solutions u s  C  C.  o f the denired  Thus  z s  C  fails  Hence, i n o r d e r t o  f o r a , we n u r t  and the orbit  lies  presune i n the  t.  Differential  in  Equations  the d i f f e r e n t i a l  z =  00  Op  that  and  f o r plane  orations  Orbits  ( p . ? ) wo new p u t -  } - ~p/f  ^ =  t h e y become  D z - 2V8U + S)Dy =  3 ( 1 + 6 f (w +  B * y 4- 2 * d 3 ( l + 5)3:/ =  3 ( 1 + S ) * ( 1 , + \t  Z  where  w =  1  z  X  j - (~275t  3  =—^(0-^+ 48  +.....  .  )  9y)  ll/SSy*)  - 25/23**7 + I S S x y * +  23/SS^y  .)  t  f  I , — i(/83x +  l:(3s+/23y)  (  j$f C ? ^ 2 S ^ - I S x y -  Ts =  p^e + X £ +  +  15/22y* )  S?y ) z  -~- ( - 2 5 / 2 S / +- 3 6 v s y + 1 3 5 / 2 3 * y*- + 2  fy ) ?  Because t h e o r b i t s a r e t o be p l a n e a n d p e r i o d i c ,  *e can  assume t h a t put  they w i l l  y ( C ) = Q.  ( p . 6 ) , wo  Also,  can p u t  cross the z - a x i s .  ¥o can  therefore  owing t o t h e s c a l e  factor'  introduced  ::(C) —  I  without  loos  of generality.  Hence g e n e r a l i n i t i a l x(C)  =• oe  y(0)  =  =  x  1  . So  = \+  a s s i s t i n the  J+  Ihe and  the  in  o ^  o ^  We \  , \,  Using in  S  f t o  r  =r  +  » conditions  will  now I0  the  > \, 0  be  definitions  ~(2V2D -  of  y23)y„ -j/S3)~  and  +.  are  c  ac  £. (p.lC)  above  coefficients:,  obtained  f r o m pajxee  since  and  d  .  w  l4  c,  &,  are  11-1P linear  to represent  the  the  integration-  . The  terms  needed.  aa :  5z„ +  integrated  Conveniently.  with  m  5  :r &+ y e  of. t h e  the  sore  »  be  «<, o / , S,  Further,,  retain  continue  «v  ae:  =« ^  7 =  t,  y  z  conditions  7  S  e * , - l , o/ =Q. we  s  tahen  i n t e g r a t i o n ' v e have put  terms w i t h o u t  putting  initial  \t  initial  by  5  e q u a t i o n s can  i n the parameters  %  0  By(0)  differential  potrer s e r i e s  c a n be  D:-:(C) =  ^ = 0  She  s  conditions  9v  p a g e 1 1 , vre may  write  the-  terns  sc Putting  *  In t h e v a l u e r  found f o r a , ?  vre o e t  l ^ s l n T + B d c o o T + T ( ^ e I n T +- i P c c o T )  where A  = - ^  +  ^ c - i ^ a  4  A'  11  u  4-  2.  / o  u  11  Particular can  3/2-  '»  2.  / o  =  integrals  bo represented  '<  (D\0  whero  the right  'io f i n d  11  11  TT  ouch as t h e above  thus:  . _  7  —  r"  ll  (P^O 2  X  H  =  (D  P'  «  CD  Z  1  ™|f- +  (S/SD 4 ^ ) g -  -f>E;  (3/2D  -  the- a c t u a l o p e r a t i o n s ? r e p r e s e n t e d on  TP  o f equations  p  -r  2^25 a  IT  n r ^  w  c +  -  above  hand s i d e s o f t h e e q u a t i o n s  4  =  J, o i n T + K, c o r T + T ( b  P  =  J'sinT + other  oiotilar  performed  containing  o i n r -t- If, c o s T ) terns  ^'  being  hero that •  Q  3 ) f  f, f t  Hence? t h e s o l u t i o n  \ =  a , ^ i n r - t - t ,ec-G.T+T(© .sinr+ |(  -  (-J,  P i n T  s=. o i t s i l a r termr. w i t h  7  With  d eosr)  te  +  K  (  priced  w  COE  T  }  constants.  theapplication o f the i n i t i a l  c o n d i t i o n s vie  find  a  . /4_4. ... 7 J 2.2. H  =  d  a* _ „ 7 A _ V 2 **» ~ ~g~ 2Z  c  6  =  r  d' = - a '  +  P r o c t h e c o e f f i c i e n t s o f S*" vre  obtain:  ^ = 5 ^  M  +  (Sy3D + V 2 3 ) y + | ^  ,P = - ( 2 / 3 D - y 3 3 ) s 2o  +  g  y  +  +  ^5 ,  > u t i n t h e 3en own yaT'iea:  7  +  9  w  _/!l i 11 r  i  &  =  cijrT + 3 COET'+TCC  s i n r D  cost )  + T (,E S l l l T + W COB T  •ifiiere _ 5/4L>  h  3/4C  3/z ~8~  ~2zT  8  3. 2  E  /4G 4  , /4t „ 11  25" 7r  6  "H * • —  14/z  i? = 8  to find  2/23 II  L =  ^7/4G Z.  c M  c  _  1 2  II  P* _ _ 12/23. _ II j  ii""  14/2 7l  "  , u  integral  wo  COOT)  .  2  2-  3/Z3  that  + T "(I* o i n T + P c o o T )  whore  + 2/23  r  theparticular  ~ i n T + K cooV+-<r(L c i n r + U  J  /2  5/2 ~4~  "It™  If  2/44  p  W O proceed  diroover  =  if  c +  D* -  1  ii  C»  ~iT"  y g 3  /35 +  lo  vg5  2572. ~22~  ~  ,, 2 ^ 6 11  43/2.H il  c  , Jo4 . II  p* _ •( p. ^  -  <•»  p  K ' = -Z/Z3 + I 2 ^ ?  ii = -6  a -  i2.c  x: •  » | t  C  +  4 i ^  d  We  • _  obtain solutions  i n t h e form  a ( r i n r - r c o s T ) + *r Ch r i n r +-  c, c c r T )  z  + V.Ccl sinY-+j ='  e, C O S T )  cirrPLlar t e r s e  zo  tahere 3/44  b  =  zo  d  /46  „  l&k Ifc  /4fe 44  _7 22-  .  n  _  ^  d  -&c  d' =  =  --Id  *»* = ^  The  general  period  conditions that  _  19/Z16  t  „  /Z3  3/Z 4-4  Q  ,/33 ZZ  4.  #  c+  ^  d  /Z3  zz  11  d  y, s h a l l be p e r i o d i c with  2af are-  x f s i r ) - %{o) ) -  s  3/Z  J/Z  zo  =  zo  «  y ( C ) ~  As b e f o r e is  +  r l  ^  e  j{lK  c  „. J7  DX(2TT ) -  Dx(r) «  0  r  Dy(S-ir)  Dy(0) =  C  ( p . 1 3 ) i t c a n be shown  a consequence  integral (p.4)  0  of the other three  -  that the last  by v i r t u e  of there  ef Jaeobi's  *!!he s u f f i c i e n t shall  c o n d i t i o n s , t h e r e f o r e , -that  be Periodic with p e r i o d  7 =  y.,+  +  trhere v/o  0  S  0  y(2-ir) - y ( 0 ) =  C  -s- a e +,  y,„<5 +  2^,  conditions to  „  y e 0 (  % , 7^ , y O  , x,  w  y^ , have t h e f o r m s  a  already  found  obtain:  =  e  - = f  T  y  C  s(3TT ) - v ( C ) =  I f '/re a p p l y t h e p e r i o d i c i t y  y.  and  2Tf a r e  Ds(2-ir ) - D a ( 0 ) =  :-: =  x  R,d )<f + S e t ( I , + T c + % d ) 5 + •  + (R,+ R c +  v  z  z  d + ( i y + R'c +  - f O +<rd+  R;d)5  ( R , % R?c +  r  -  ~  r  3  1! =r — -i- Taenia TT  By  !Ef> +  R/d)^ + G £ + ( ^ + ^ c + B  r  T j d ) ^  t-  8  u  T" — 3/44 _,. 3-n-  R  '  3  ~  = TT  m«  the theory o f i m p l i c i t f u n c t i o n s ,  7 / ?  .  T ^ ' d ) ^ +• .  II  '  5  + P,V +  y-^^  the f i r s t  two a f  the  above  terras  throe  of £  equations  and  eliminate  c  c = nj  o)  £ .  and  can be s o l v e d f o r  d  from  - pR  /o  the third equation  Tn ^ a r t i c u l a r , The  obtain  I, =  If  a n d one  7  of  cane, the oorieo  f o"H  =• 0,  <5 e  +  t  0*70  the i n i t i a l  initial will  may ^ /  A  C,  there  be w r i t t e n  c  and  £^:  in i f £;  Whatever  <T  the i n i t i a l  n a y b e e a p r e e o e d ao  these  the  bo a - e r i o d i c .  for S  a r e two s e r i e s i n  d  conditiono and  uiii  .  +  and  hove  ;  as  r,  y  in  r o r i o a i n . €,\  of  and  + i  C.  a r c two oortoo  Inaemuch ao  conditiono  H;R;  t h e r e f o r e be s o l v e d  there are w o  also that  r e r i o o i n eS.  ^  -  U •*  there  tt, , b*  the  +  0  -  Pi, can  P, 7^  0* =  t  »  -  0 = 0 ;  if  S = S  U  :  e%  conation i n  of-  where  to  cR" •+ H "  -  ;  -  of  in  P v means o f t h e s e c c ^ u t i c r . r a o  O  power  d  u  0 =  tt  and  + o £ + ^i/-* i3 §e -t.  whore  terras  c  valuer  ore l i n e a r y  -ne ir<ay  functions:  arouste  a r e chooon 00 t h a t x ( 0 ) ^ l  s  y(C)=C  t  z end  Tn t h e d i f f e r e n t i a l They  equations, r u t e  - 2 ^ 2 ( 1 + 5 ) D y = 2 ( 1 + £ f (X, +  Dy  - 2V3(l + $)Dx - 3 ( 1 + 6 f ( I , +  2  Substitute  y = 7c +  £ =  ;  )  initial  follows:  conditions  Wo  a n d §. a r c t e b e c h o s e n t o  periodic.  c©~ditionc  x,(f) » i "=  that  0  the s o l u t i o n s  ssero '"hen  of  ?  :  c e n d i t i --nr a r e t o b e  y(f ) =  T.hs i n i t i a l as  4  2  i  . . i. .  z  initial  other  nsbe  X £ '+ I £ -t- . . . . . )  i n then",  y,£ -+ j t~+.  =. 1 ;  The  £ .  £ +- <5", + .- - The  x(0)  I T  become:  Dx 2  l  1,  affect  the series  a l lt h e other  for  coefficients  s,  y, are  s.  nca proceed i * i t h  the existence  proof.  the integration  after  t h e manner  germs hoi  Withj^t^e. "C^b© t h a t ^ a r t o f  A. h o m i n g t h e c o e f f i c i e n t  after" the s u b s t i t u t i o n of t h eseries  f o r - ap y .  e'  Then  define: £ s  D %  g  D^y. + S/SDs. - 3 1 ^ *  =  5!he t e r m s w i t h o u t  - 2/SDyv  e  arc  - 2 1 °  therefore:  e  'fhere a r e r e a d i l y i n t e g r a t e d  3  »  —  8  .sin.T+ c o e r  Here' we f i r s t  y„ = °  8  einr  make- t h e d e f i n i t i o n s :  !  • H e n c e , when differential  to give:  t h es e r i e s f o r  equations,  4.7  S  s  s  ~  y, a r ep u t i n the  t h e c o e f f i c i e n t s o f the t o r s o  0.1*6  i/hen  we f i n d *.  thevalues  of  a  y , a r ep u t i n t h e above,  i n e  -4-  Urinr  t h e R o t a t i o n o f pcoe- 3 0 , '.TO c a n r e p r e s e n t t h e  particular  integral?  Here ,  j ;=  Hence, vrith of  initial  Terms  i-n g  of the above  -  differential  the proper i n t e g r a t i o n ,  conditions  obttin  in  £  2,  i n bno-m v a l u e r  2  vo  and t h e  the periodic  r  Putting  equations  0  The c o e f f i c i e n t s o f t h e t e r m s  x  c.  4"  obtain  application  delations:  ^  _ tQ3/<& 3C3Z>  -  Z2  ©/  n  ^ 1  = 23/138. 3C3Z>  Z  _ 31/6^ 3C44>  Z  &' = — * 27  l  ^ 5<>Z-> Z  A  Let us now c o n s i d e r t h e p a r t i c u l a r i n t e g r a l s . , Y „ a r e found f i r s t .  fauctions fl =  %  .sinjT + D-Coajr) ;  The  Ehejr a r c :  £ = ^ ( a t sinjT +  falcosiT)  ~n here b  . 55&2 3CIZ&)  = 20  e  =  2/  - 4^5" Z. » 22-  3C3Z)  V  r£f£|  =  0 _ ~ z~  a*  = (const.)<C  I3/6/Z3 TC3Z)  Z  b « _4 2J  ^ '  IZ8 b\ =  (conet)^  . , _ 79/3 " 3028) z i  Ovrlng t o the t e r s e i n 1 © ! T , the p a r t i c u l a r i n t e g r a l s w i l l contain can  terns  i n T*"£?$T •  She only -s'ay -.JO  a v o i d t h e s o non-periodic terms i s t o tnalce  W i t h t h i s done, ore proceed * * i t h the i n t e g r a t i o n t o obtain  t h e oonp'ieto e o l a t i o n i n t h e f e r n :  ••/her  27CIZB)  2 1  8  z  7^""  -21  2.1  -  247/4/Z3 E7CIZ8)  /^C 8  r  83/3  ProR  the coefficients  •".nations: the  solutions  Une  terms  ^ =  ^  of £ A  we ;  obtain g, =  the  differential  6, A-  f  of which are:'  4*  in £  give:  J-I  "}g  =  frors  rissilBi* e x p r e s s i o n which:  Tith  Y* o  and  Dx' c  J  ^s  Z  4  * .feinilar expression  with  primed  constants  '.?aere  27(256)  tf'  =  K/2Sd + ? d » M  - .5.Q44D/Z3  1*1© 11. c o  s » £ >  vrhero  >i•a \ ssxinn; ji rT + + bO 'P cos.j T ) a  =  a' =  (-ii<*  ~ %/2/b' +  l'-.I<^ + 2/3/5, +  41  fed  /  Z.P-41  41  )  i P *  As b e f o r e , i t i s n e c e e r a r y coc T  &Zoi' )  t o aahe t h e c o e f f i c i e n t s  of t h e t e r m s  i n  vanish.  From t h e o::io|once p r o o f ,  vro hnsor t h a t  t h e same v a l u e o f $  z  vill  oaho a l l f o a r o f  the  aboyo coc-ff i c i e n t o  valuer will  of  c ^  j  3  ^  (  vanish, and ^  S  antowetically nate  are  only  Hhe  one  teo nay  She  sd  "de  , t h a t r-alo (  , bh  (  independent equations b o n r - e d as  values of  <5  a c h e c k on re  t  notebcsrover,  fonnd and  b  , vanish.  East  the  , vanish,  4 /  Hence  that the  fchet  there  satisfy.  other.  checked/ a r e :  72  T n a p f i t i p n t p the, G o n o r a l , m  Since to  the  there  gem  , V',  e x p r e s s ions? f o r for  £. , P.  , x. , t h e  y. , a r e  fornor  quite net  w i l l  be  analogous explicitly  m e n t i o n e d i n whet, f o l l o w s . I P -no c n b s t i t u t e t h e differentiel oi  ^"  £.  and  y=/  £  equations  series for  ( p . 2 8 ) and  %,  y,  eqttatg the  in  the  coefficients  *Te o b t a i n t  j* '  _ j=.<  -  ^  y='  Zn  + / : X X P J ;  .  J  y^/  '  Let boon  u s str---;cse no-? t h a t  found f o r i = C h  and  X  that  the v a r i o u s  a.« <$" ,  1 , ... , Sn - 1;  s  1,  = 0  r  ... , S n » 3;  that.  2«-  -  7  a  =  '; c  o  3  s  i  n  3 T +  d.  y  oos 1 T >  sinT  2w-i  2rt—/  in which c^"^  of**^ and  d^"~^ a r e l i n e a r of S  i s indcoeisdont She  terms i n  £ " 2  S  f u n c t i o n E OS  , <£ „ ;  a n d <5  ore therefore  »4L  represented  '; «<£-<5T  by  J—0 •whore  o<  a n d /3,  arc linear  in  5  Hence  j  in  —  217.  j^o  U.  •>  Shore  0  c OS4  J  ore linear  are consistent.  snieyie v o l 0 0 ;  that  b,  • .'  T  )  - S / S ^ V  a f " ^ (- ^  whore  they  0 1 0 7 T +  i nS  /§§  and bv t h e oaistonco  Hence "o c a n s o l v e  a value  the eolations '.Till  )  t h a t v ? i l l r/ahe bo P e r i o d i c .  for$  zn-z  "roof end  a ^ s b ^ s C  obtain ro  If. the i n t s r - ' a t i c n conditions  a ' r p l i e d , t h e 5-olt2ti.cn t r i l l  Co.  • 2 "  •'•here  rinjT-t- C  c '"'' a n d  —  o  H  v'here c<  Zn+i To  2n-| j-o eliminate  integration,  be  ere l i n e a r  In  21 l o f  and  ^3,  i n the  fern:  S  >i  t r e a t i n g the terinc i n  -t-<3  initial  .cor-jT)  d""^  c  Similarly  -'  i r c o r . p l o t e d e n d 'the  v 3  are  y  we  r. i T +  <-»  independent  got:  -j T )  of  J the t e r r r we  t h a t 7rt">7 be'.cotae n o n - p e r i o d i c  snot pat  & _ zn  in th  r  t  Hence  •i  —  /hero  d  c  c  a  w  ^  Ic- h a v e  CXP.T  ir. independent thup  fr,r c o n s t r u c t i n g and  that  ohovm  of $  that  'rzlne  s u c c e e d i n g r--3.ly.ee f o r S~boor i n d i c a t e d t h a t the  of  •  t h e p r o c e s s commenced  the o r b i t s can  a f t e r the  » S  <5"_  b e c a r r i e d on b a r been  (p.2?)  indefinitely;  celected  the  are unique. F u r t h e r , i t hac o f o d d s u b s c r i p t a l l vanish.  If  o r,w  ?;a?  introduced  a =  x£  7  7 e +-'  B  =  p u t bach  :: £  +  y e  (p.6),  and resionbor he"  t r e obtain f o r t l i e  solution:  -V . • -  l  z  0  £. f o r £.*" ( p . S G ) ,  4- - - -  x  t  o" = S e + .. . z  2hat  i s , i n t h e neighbourhood o f t h e p o i n t  8  30Z8)  24  L  V.  72-  I (p.4)  |8  7  27(32) *  J  "  C  9(lZ8)  3(128)  8  L  V  -  con  £  /  solutions  j  /  a  pararaotor £  C  ,  27(VZ8)  the sign  denotes  of  /3  the initio"  Using  will  conoorgo  t h o above t e r n s  r  f o r |£[  i n t h e above  displacement  from t h e L a g r a n g i a n p o i n t p a r a l l e l t o t h e abo?e s o l u t i o n s  smT  ) J  neighbourhood o f t h e p o i n t I I .  i n the  be o b t a i n e d by c h a n g i n g Tho  r  2 7 ( 3 2 )  270^)  The  36  ^"-oais.  sufficiently  a s have p l o t t e d  orbits.  1'ho  snail. I'Jhen  X  30 the valne  £ — • C - l tras u s e d  i n order  t o c©spare t h e  d i n p r - a w i t h Buck's, t h e o r b i t s a i d n o t s h e w a s She  orbits  shcrn  for  f. =  .1  l i ewithin  distinct.  the region of  „c on v e rg e u c e { p . G ) .  (1)  H t m l ton,'  ».R.;  (Hew  (S)  Bach,  S.;  MJ^£c^.cJ;XoJL^ lor!?,  1?35);  Chap.  YTTI.  Houston's P e r i ^ d f ? O r b i t o (Washington, 1930); Ohap. I X .  

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