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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. H9. . Isosce/es Tr/an^/e £to/«4.t~tor?s of f~he pn>bfem of three bodies iWo orvirhfGh are 06/ate ^pheroicjjz, Y char/otiie Is/ay Uohn A. L£3 37 J6J7 I S O S C E L E S T R I A N G L E SOLUTIONS OF THE PROBLEM OF THREE BODIES TWO OF WHICH ARE OBLATE SPHEROIDS. By Charlotte Islay Johnston A Thesis submitted for the Degree of M A S T E R O F A R T S in the Department of MATHEMATICS THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1?25. >C/. A^-e_-t__-e-^£ t Isosce les 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 inf inites imal and the two f i n i t e bodies are res tr ic ted so that they move in a c i r c l e ; in Part II 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 straight 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 Finite Bodies move in a Circle and the Third Body i s Inf ini tes imal . 2 The Differential Squation of Motion.- Let m, and mx be the two f i n i t e bodies and jx the inf initesimal body. Let the units of mass, length, 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 Prob-lems del Tre Corpl," Annali di Matematica Series I I I , rCl XIII (1907) p.p. 179-202. V. MaoMlllan - "An Integrable Case la 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 Inst i tut ion of V&shington Publi-cation 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, , ma, and >x be £ • JJ » J ; £ . 7) . J ^ d F , ??, J , respectively. At the initial time tg m, and rn^are put in motion at the points $, 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. Then Let - 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 ±(o) = /. (9) Then 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 Zo -fZ0=o. I t s solution i s z0 ~ Ao era T -t Bo ^u^o r but when (9) is satisfied it becomes The differential equation from the terms in «- is •2.(0) - 0 , ^ o '«>> = / , V - ^ ' " v^«< 2 + Z ^ -^z-A.I/^'A f3'iZ 3 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 there fore we make -2- ^./o 2. 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 respec-tively, 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 off is where (11) N) 4r , Aty •K-i 3-k-ti ' aad 6L/k 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 -J7 "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 Then £ a - J ~ ; 7, * - £ £ = ~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 concen-trated at the centre, the differential equations of motion are 1*AS 3 > <14) where / Let ^ = £ + r'r, ^ J'/f(H-T}'Tj $= 1 S„. «.V ? L 7,= r' v (15) - 6 -Then 7jt - A ^ , u-flu -/ll>i f Cf-3 zr (2 A, T>r si. ?>J XT' ~3_ /^c+..V From (15) " -/-V^r/* (16) ; -^ -vt ^ >k 1^ + 2 A, V- - 0} For u,^ <? the solutions of (16) are (17) y^lYiOiTTr, ; ' - * • l e t ^ft/ifrl ^-'J^oTrTr-i-^o, }--i/+ur; (18) where ^ r_ ^  - 6^ = o when ^-- o, Wh#a (18) is substituted in (17) the differential equations for /t, T^ and ur beoome •*vO+fi)z (i+^tv The second equation of (18) i s sa t i s f i ed by (19) where (20) <*• = <*-t+X jA-t (^ -a. < V > ^ 4 ^ J ; ^ = Jfa~0+X) • - 7 -When (19) is substituted in (18) we obtain ur - right member of the third equation of (19) . 3 Construction of the Periodic Solution of (20) - Let (21) / • = • ' • ' (22) and let JL = f<L, as in Part I. When we substitute (22) in (21), expand, and equate the co-efficients of fjL t we obtain differential equations for /^ - and "A W•* ]?or the terms in jx. alone these are (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 -t- /3 ^^->J7<V, Pf contains (/> to even degrees only, and is therefore a power series in cu with cosines of even multiples of/ in the co-- 8 -efficients, 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 are oonstants. If f£ 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 there-fore 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 non-periodic 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), (24) When/2, is determined W, is known. 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 ^yA 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, &>« /, £7. '*>> = 0, (27) eince dk+f) A n 'f 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 so lu t ion a / whose i n i t i a l values are £rx(c)^ o, ur-^ _ if (29) l e t us- - ^ - = X -t AT <9t (30) where , Since X - ~k (%^-) each X i s a sum of s e r i e s of odd mul-t i p l e s o f r , the h ighes t mul t ip le being y'+t . Since the ©o«ffiatents of <VaJ and <P • are equal in magnitude and s ign , v v and ainoe the sine of the highest multiple of T in X, is v 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) (>) o) J where "^  and ^  are constants of integration. To obtain the solution of the se d equation of (23) we vary the param ters 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 (b) */''<¥ +*i™fXi-A(<p+T4)}*W,. (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 (34) where ^/ ' ; and %''yare cons tants of i n t e g r a t i o n . ViThen the i n t e -g r a t i ons are performed in (34) the so lu t ion becomes ^ = ?/*<PtyJ0fX+Arq}i-S.- ccM/^rCf. . ( 3 5 ) Sinoe the c o e f f i c i e n t s of s i n ( jy> / )T and cos (y> / ) r in XAJ and ^ r e s p e c t i v e l y are equal , the terms containing the s ines of the h ighes t mul t ip les of T" obtained from the t h i rd and f i f t h terms of the r i g h t member of (34) cance l . Sf i s t h e r e -3 F. R. Moulton - "Periodic Orbits," Carnegie Institution of Washington, Publ icat ion No. 161, p . p . 21-23. * 12 -t M n ft power s e r i e s in a?*'(.•- 0/ .. -©^ mul t ip l i ed by a sum of s ines of odd mul t ip les of r , the h ighes t mul t ip le being *y>' . ^ i s a power s e r i e s in a, • from (21) ur(6)r.o and therefore 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 a 5 C U<t*>} 2J-H J (37) 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^ ( 3 8 ) ffee complementary function of the first equation of (38) is P4 Is composed of terms froa fyj p^ g rt,, multiplied by even powers of </';"^ multiplied by odd powers of </ j and a constant term. It is therefore a triply even power series multiplied °y i- • If /* is not an integer we make Ax--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 is^yv/ . 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 ur7-are 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 ur -J = TL, (7L)q t -ns[z)(X+A TQ} (40) mstants of integratior solution has the same form as (3.5) namely where ri, and \ are cons ion. The general - 14 -where Sx and /Vl^have the same form as W2 and M, " respect ive ly . Since u^ceJ- o ^ then 7]'x)= o ancl in order t ha t ur_ s ha l l be per iod ic A " a x ' a3 where P CCL)1B a power s e r i e s in cv\ The so lu t ion of (41) i s there fore 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 of^, 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 C3 yis a sum of cosines of even multiples off, the highest of which is 2/. Suppose that /*•. un >? u) r>0) /,-•) . . . r-/) have been ob-talned ant that <•*> /v. - -ha IC^'aS (45) - 1 5 - • (o) ^ fl-, ^ t > > > , . (45) where the C.v S//f/ P£V«> have the same form as the C>/y}, ^,'^'1 P10^ 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 ^ # Hence all the terms in P, are known. The terms independent of ^  are a triply even series multiplied by -fa . The terms in "f are i(y-^ - — ~ — y 3.S Tn- is odd or Q.i/e-n, multiplied by yKJk^o, •••**),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 K-- a. AC = Z CL- J - o 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 bc * , (c-\ •-•—)• (48) When (47) and (48) are substituted in (22) the values of h, and ur become (49) /=« ' ***. a?*-* j*i *- > 7t=/ '*' /=: 0 From (20) - 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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