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Isosceles triangle solutions of the problem of three bodies two of which are oblate spheroids Johnston, Charlotte Islay 1925

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U.B.C LIBRARY  *0C. H 9 . .  Isosce/es Tr/an^/e £to/«4.t~tor?s of f~he pn>bfem iWo orvirhfGh  char/otiie  are 06/ate  Y Is/ay  A.  L£3 37 J6J7  of three  Uohn  bodies  ^pheroicjjz,  ISOSCELES  TRIANGLE  SOLUTIONS OF THE PROBLEM OF THREE BODIES TWO OF WHICH ARE OBLATE SPHEROIDS. By Charlotte Islay Johnston  A Thesis submitted for the Degree of MASTER  OF  ARTS  in the Department of MATHEMATICS  THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1?25.  >C/. A^-e_-t__-e-^£  t  I s o s c e l e s Triangle Solutions of the Problem of Three Bodies, Two of which are Oblate Spheroids.  1  Introduction."  Periodic solutions of the problem of  three bodies, when two of the bodies are oblate spheroids of equal mass, are herein considered.  Two oases are discussed:  in Part I the third body i s i n f i n i t e s i m a l and the two f i n i t e bodies are r e s t r i c t e d so that they move in a c i r c l e ; i n Part I I the third body i s a f i n i t e sphere and the other bodies move i n i t i a l l y in a c i r c l e .  In each case the orbit of the third  body i s a s t r a i g h t l i n e perpendicular to the plane of i n i t i a l motion of the other two bodies and equidistant from them. fhen the two bodies are spheres t h i s problem has already been discussed.1 I Periodic Orbits when the F i n i t e Bodies move in a Circle and the Third Body i s I n f i n i t e s i m a l . 2  The D i f f e r e n t i a l Squation of Motion.- Let m, and mx be  the two f i n i t e bodies and jx the infinitesimal body.  Let the  u n i t s of mass, l e n g t h , and time be such that the mass rai-mi=-^-, the distance between m( and mt and the gravitational constant 1  Pavanini - "Sopra una Huova Categoria dl Soluzionl PeriodIche nel Problems del Tre Corpl," Annali di Matematica Series I I I , rCl XIII (1907) p.p. 179-202. V. MaoMlllan - "An Integrable Case l a the Restricted Problem of Three Bodies, M Astronomical Journal, Nos. 625-626 (1911) D. Buchanan - "Isosceles-Triangle Solutions of the Problem of Three Bodies, 1 * Periodic Orbits, Carnegie I n s t i t u t i o n of V&shington Publication Ho. 161, p.p. 325-326.  - 2 shall both be unity. let the origin of coordinates be the oentre of mass and let the J7-plane be the plane of motion of the finite bodies. Let the coordinates of m, , m a , and >x be £ • JJ » J ; £ . 7) . J ^ d time tg  F , ??, J , respectively.  m, and rn^are put in motion at the points $,  At the initial 0, 0 and  -i, 0, 0 respectively, so that they move in a oirole.  Then  The differential equation of motion for JM is 41S - -_JL  fli-JL &\jt?(H-±£)  t ••• ' V ^ ' ^  U*VJtl~ \  (T\  where t£- is the minor semi-axis and -e the eccentricity of a meridian section of m and m. 3  Oonstruotion of the Periodic Solution of (1) - By expan-  sion (l) becomes  where &)-- ('MS) •• • (-*&) Let J-A-zand auppose^^<r«, where <T and ^ are constants.  Then  (2) becomes  Let  t-t, = JfrTJTJJr  (4)  where T is the new independent variable and ST is a constant to be determined.  Let  Then  - J -  z - 1 z^.o-V  (7)  where the £" • are constants to be so determined that the z will be periodic with the period -z-77 in T.  At T-<? let  ; =~.  7=*,  (8)  Then }= <*1. <£? ^ <^J Mo+ S) or CL, = F/g[7^YJ 41  . Thus ^ is  proportional to the initial projection of the infinitesimal from the plane of motion of the finite bodies. Prom (8) we obtain •2.(0)  -  ±(o) = /.  0,  (9)  Then  V- ^ ' " ^o  '«>> =  /,  v^«<  On aaostituting (6) and (7) in (j>) and equating the coeffioi•ata of the powers of «^we obtain differential equations for the z,. She differential equation from the terms independent V of <*/la Z o -fZ 0 =o.  I t s solution i s z 0  ~ Ao era T -t Bo ^u^o r  but when (9) is satisfied it becomes  The differential equation from the terms in «- is 2 + Z ^ -^z-A.I/^'A f3'iZ3 1> to  2- -^^  JZ.  Sin r in the right member gives rise to a term of the type reoar in the solution ofz^. Since a periodic solution is required this term in alnr must not oocur in the right member  - 4 and t h e r e f o r e we make -2-  2.  ^./o  The general solution for z is -Z. /i>  and when (9) is satisfied it becomes  It may be shown by an induction to the general term that the S" • and the arbitrary constants of integration are uniquely determined by the periodicity and initial conditions respectively, and that each z. is a sum of sines of odd multiples of J  T, the highest multiple beingy'-t/  . Hence-the periodic  solution of (l) in terms o f f is  (11) where N)  4r , •K-i  aad 6L/k  Aty 3-k-ti  '  is a constant, containing /"in general.  The period  in -r is 277 and in t it is  JY  y./o  Periodic Orbits when the Third Body is Finite 4  The Differential Equations of Motion.- let in, m^ repre-  sent the oblate spheroids as in Part I and let f>c denote the third body which is here assumed to be a finite sphere. Let the masses in and m^, the initial distance between them, the gravitational constant, the origin of coordinates, and the  - 5 J 7 "Pi®11® b® *n® same as in Part I. Let the coordinates of m,, m^ and /^ be £ , ^ , £ ; £ , £ , £ and J f ^ , r respectively. At the initial time ^ let  £ a - J ~ ; 7, * - £  Then  £ = ~k >  l^%  > I -h •  (12)  From the centre of gravity equations, which are  and from (12) it follows that  When it is assumed that each oblate spheroid attracts the other as if the mass of the attracted spheroid were concentrated at the centre, the differential equations of motion are  <14)  3 >  1*AS  /  where  ^ = £+ ?  Let  L  7,= r  ' r , ^ J'/f(H-T}'Tj  $= 1 S„. «. V r'  v  (15)  - 6 Then XT' ~3_  7jt -  A ^ , u-flu -/ll>i  (16)  f Cf-3 zr (2 A, T>r si. ?>J /^c+..V  From (15)  "  -/-V^r/*  ;  -^ -vt ^  (17)  >k 1^ + 2 A, V- - 0}  For u,^<? the solutions of (16) are  y^lYiOiTTr,  ;  '  *  -  •  let  ^ft/ifrl  ^-'J^oTrTr-i-^o,  where ^ r_ ^ - 6^ = o  when ^--  }--i/+ur;  (18)  o,  Wh#a (18) is substituted in (17) the differential equations for /t, T^ and ur beoome •*vO+fi)z  (i+^tv  (19)  The second equation of (18) i s s a t i s f i e d by (20) where <*• = <*-t+X jA-t (^ -a. < V > ^ 4 ^ J ;  ^  = Jfa~0+X) •  - 7When (19) is substituted in (18) we obtain  (21)  ur - right member of the third equation of (19) . 3  Construction of the Periodic Solution of (20) - Let (22) /  •  =  •  '  •  '  and let JL = f<L, as in Part I. When we substitute (22) in (21), expand, and equate the coefficients of fjL t we obtain differential equations for /^- and W•*  ]?or the terms i n jx. alone these are  "A (23)  where  0,= a sum of cosines of even multiples of r , the highest </  •  multiple being zj . The oomplementary function of the first equation of (23) is fvt - A , c^oJ~7<T  Pf  -t- /3 ^^->J7<V,  contains (/> to even degrees only, and is therefore a power  series in cu with cosines of even multiples o f / in the co-  - 8efficients, the highest multiple of r in the coefficient of a* being *.«: •  Suoh a Beries is called a triply even series,  fhe particular Integral of the equation will also be a triply even series. Let it be Cjr) where each term is of the form  where c- c^} •• - c/'J If f£  are oonstants.  is not an integer the complementary function does not  have the period SJT ,  in order that this condition of period!-  oity may be fulfilled we must make  A,-e,  = o;  and the solution then is  A, = c, (r) . If /^ is an even integerP will contain cos//Tr and therefore the particular integral will contain terms of the type fsin-r , but no arbitrary constant which can be used to annul •uoh terms. Since the solution is to be periodic-«/* must not be an even integer. It will later be shown that when J7< is an odd integer nonperiodic terms arise in the solution of foz. Srcluding the case when JK is an integer, even or odd, we obtain the solution of the first equation of (23) in the form  A, - Cjr), When/2, is determined W, is known.  (24) Since it is composed of  terms in odd powers of ip and terms in/i  multiplied by odd  powers of ^ , it is therefore a triply odd power series. If we neglect the right member of the second equation of (23) we obtain  - 9 -  ^T -tflt  S OyCO^]ur  -- Gj  (25)  whieh is one of the equations of variation, iff being the generating solution.  Sinoe <f is a function of j==S-  , con-  taining two arbitrary constants to and a,, the tv/o fundamental solutions of (25) are obtained by taking the first partial deriTatives of v with respect to t0 and a,.2  One solution is  whioh is periodic with the period -27T in -r. Let the factor "~ JjSrSj**9  al>80rlt)ed  °y tlle undetermined constant by which the  solution is later multiplied.  The solution is then  ur = 0 = 2 ^ y A V = C^r-t-3. a^cnrsr-c^xrJ'-t  •• <  (26)  where  and the $? ere sums of cosines of odd multiples of/. Whenr=c? K , &>« /, eince  dk+f)  A n 'f  £7. '*>> = 0,  (27)  in equation (10).  O B differentiating y with respect to <a. we obtain the second solution whioh is us  ^ (l#)  + \J*M  .  (28)  % Poinc&re - "Les Methodes Houvelles de la Meoanique Celeste," vol. I, chap. 4* p.p. 162-232.  - 10 Therefore fh« initial raluea of this solution, from (27), are us~tx {a) =. o;  In order to hare a s o l u t i o n a / whose i n i t i a l values a r e £rx(c)^ o,  ur-^ _ if  (29)  let  us- - ^ -  = X -t AT <9t  (30)  where  Since  ,  X - ~k (%^-)  tiples o f r ,  each X i s a sum of s e r i e s of odd mul-  the h i g h e s t m u l t i p l e being y'+t .  ©o«ffiatents of <VaJ and v  Since the  <P • are equal in magnitude and s i g n , v  and ainoe the sine of the highest multiple of T in X, v  is  obtained by multiplying the sine of the highest multiple of f in(V^.by 1, being the term in -^ which is independent of «% therefore the coefficients of the sines and cosines of the highest multiples of r in X. and Q . respectively are equal numerically and in sign. The general solution of (23) is therefore ^~~ Kf + ^fX+ATQ'} (>)  (31)  J  o)  solution of ^the equation of (23) we vary the parameters where "^ and aresecond constants of integration. To obtain the 7v0)t and ru'2 in the following way: (•)•  nvf"<9+*>l'}(X  + ArQ},sa'.-  (32)  - IX -  or, - ^q+yll'tf-JC  + AfQ + rQjji-^Q  + ^{X  + Ahq-t-rq)}.  When the above values of «V and ur are substituted in the second equation of (23) we obtain  When ?z/''and •n/'^are constants (31) satisfies (25) and  fharefore  */''<¥ +*i™fXi-A(<p+T4)}*W,.  (b)  (32)  She determinant of the coefficients of ?i/';and ^frora (32) Is a constant^ and by (27) and (29) it is equal to 1.  The  solutions for Tz/'-iand Tt^ara y  y  (33)  Shan ?z, and n, are s u b s t i t u t e d in ( 3 l ) we obtain *> = ^q+^fX+Arq}^  QfvtXe/r+ApfftfWyr  + Xfw, 4 V r  where ^/' ; and %'' y are c o n s t a n t s of i n t e g r a t i o n .  (34)  ViThen the i n t e -  g r a t i o n s a r e performed in (34) the s o l u t i o n becomes ^  = ?/*<PtyJ0fX+Arq}i-S.-  ccM/^rCf.  Sinoe the c o e f f i c i e n t s of s i n ( j y > / )T and c o s ( y > /  .(35) )r  i n XAJ  and ^ r e s p e c t i v e l y are e q u a l , the terms containing the s i n e s of the h i g h e s t m u l t i p l e s of T" obtained from the t h i r d and f i f t h terms of the r i g h t member of (34) c a n c e l .  Sf i s t h e r e -  3 F. R. Moulton - "Periodic Orbits," Carnegie Institution of Washington, P u b l i c a t i o n No. 161, p . p . 21-23.  * 12 t M n ft power s e r i e s in  a?*'(.•- 0/ .. -©^ m u l t i p l i e d by a  sum of s i n e s of odd m u l t i p l e s of r , the h i g h e s t m u l t i p l e being *y>' .  ^  i s a power s e r i e s i n a, •  from (21) ur(6)r.o and t h e r e f o r e a / ^ / U o, <Y-< • • • - ^  ^6)  from which it follows that # o _ o In order that urt shall be periodic the coefficients of r in (35) must equal tero, and therefore [  A  «-  when P*'Wvia a power series in a1 with constant coefficients. When these values of 7/" and 7/''are substituted in (3J>) the solution becomes 0^1  5  a  C U<t*>}  2J-H  (37)  J  where S,rv*ks a sum of sines of odd multiples of r , the highest multiple being */•' • The differential equations for the terms in /^"are K  + K K ^p- j  urxt (it %eO^^}urt  -\N^  (38)  ffee complementary function of the first equation of (38) is  P4  Is composed of terms froafyjp^g rt,, multiplied by even  powers of </';"^multiplied by odd powers of </ j and a constant term.  °y  It is therefore a triply even power series multiplied  i- • If /* is not an integer we make  A  x--f3z^o  in order that the  solution may be periodic with the period m in r . If Ik  is an even integer it has been shown that the solution  - 15 of A is not periodic, therefore this case must be excluded. If JK is an odd integer  where 5 J^'is a sum of sines of even multiples of r , the ok  highest of which i s ^ y v / . When urt is multiplied by odd powers of ^ in /f the-5! give rise to terms containing sums of cosines of odd multiples of r.  Since JK is odd there will  be nonperiodic terms of the type -Tain/xT  in the particular  integral. The case where A" is an odd integer must also be excluded. When JK is not an integer  A ^ ii 2 C^«.V  (39)  where C 2 ' is a sum of cosines of even multiples of r the higheat being -y'. The terms in W; are multiples of /^,/^ «/AJ» ^T* 6 ^ and powers of ^, combined so as to give triply odd series. Those terms involving/^ and <^fare multiplied by JL_ and those involving ur7are multiplied by •£, • The lowest power to which a. enters «-'*£ 2.  is found from the term in o/r^to be 3. Therefore where each W^  is a sum of sines of odd multiples of f the  highest of which is zj+i. The complementary function of the second equation of (38) is urJ =- TL, (7L)q t -ns[z)(X+A TQ} where ri, and \ are constants The general mstants of integration. integratior solution has the same form as (3.5) namely  (40)  - 14 -  where Sx and /Vl^have the same form as W2 and M, " r e s p e c t i v e l y . Since u^ceJ- o ^then  7]'x)= o ancl in order t h a t ur_ s h a l l be  periodic A " a x ' a3 where P CCL)1B a power s e r i e s in cv\ The s o l u t i o n of (41) i s therefore ur  , J~  2 S/^V/"  (42)  where the \ have the same form as the ->, in (37). Consider the terms in p3 of the first equation of (20). The differential equation for h,3 is f3  + Kfa  ~~P3.  (43)  Since J7< is assumed not to be integral, the complementary function of (43) is not periodic and therefore must equal zero. She terms in Pi are combinations of multiples offh^w^a3^  uf  "^a^and powers o f ^ , combined so as to give triply even series. Those terras involving/^ and ur are multiplied by-^. Those involving ur and ur are multiplied by -^-y. The lowest power to which <=t enters ^ i s found in the terms in ur^ and ur to be 2. Hence  to-±, 5 C , V  (44)  where C3y is a sum of cosines of even multiples o f f , the highest of which is 2/. Suppose that /*•. un >? u) r>0) /,-•) . . . r-/)  have been ob-  talned ant that  <•*>  /v. - -ha IC^'aS  (45)  -15  (o)  ^  where the C.v C>/y}, ^,'^'1  S//f/  P10^  fl-,  -•  ^  t  > > > ,  .  (45)  P£V«> have the same form as the  respectively.  The differential equation for  the terms in JX in (20) are  P  containsjv.fUr)aniL$  since the terms of the first equation of  (20) containing"^ also contain JA . ^ H o c c u r . i n ^ # the terms in P, are known.  Hence all  The terms independent of ^ are a  triply even series multiplied by -fa i(y-^ multiplied by yKJk^o,  . The terms in "f are — ~ — y 3.S Tn- is  odd  or  Q.i/e-n,  •••**),where * is odd or even^respectively,  and these terms therefore form a triply even series. They are multiplied by -^_0  since ur.y appears among them, and therefore  the lowest power of <=t in a* ^ is 2. We thus obtain fhe particular integral only is periodic with the period i-TT. fhe solution is therefore  fhms by induction  ^ -- ^fe* S C j v ; ^ ^,4j - • • oo;.  (47}  When pur is known all the terras in n^ are known since only /v. (J < >")_,  and ^,(j*Lir)  occur inlV^. The terms independent of  ccare triply odd power series multiplied by ~^rc^-,) since /i^ occurs among them.  The terms in ur-1 Q^r^axe  multiplied by ij>  where K is even or odd as m is odd or even respectively. They form a triply odd series multiplied by -^> . The lowest power to which «- is raised in CLWappears  in the terms ^ _ / ,  - 16 .<^^and is therefore 3« Hence  ^--rfe2 W **"«.*" '*.- a_~ where wJ*J+'}  has the form of W/V"^  The complementary function of the second of equations (46) is  The general solution is <*;  where (7  and ,V  AC  K--  = Z- o  a.  J  CL-  Tj, and ^ are constants of integration. In order that urf0)~o and that the solution he periodic with the period Xilj  7,  7*  ~ °,  4 ^  *^  "*  from the results obtained above  By induction ^  = ^c p0  b  c  *  , (c-\  •-•—)•  (48)  When (47) and (48) are substituted in (22) the values of h, and ur become /=«  ***. a?*-* j*i  '  *-  >  (49) 7t=/ '*'  From (20)  /=: 0  - 17 (4?) and (50), together with (12), (l3) and (18), give  J - 1 - °> which are periodic solutions for the orbits of the three bodies, two of which are oblate spheroids.  BIBLIOGRAPHY  Poinoare, H.  Lea Methodes Houvellea de la Mecanique Celeste, Paris, 1892.  Annali di Matematioa, Series III, vol XIII, 1907. Astronomical Journal, Bos. 62j> - 626, 1911. Carnegie Institution of Washington, Publication No.l6l, 1920.  


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