Oscillations Near an Isosceles-Triangle Solution of the Problem of Three Bodies as the Finite Masses...

Oscillations Near an Isosceles-Triangle Solution of the Problem of Three Bodies as the FiniteMasses Become UnequalAuthor(s): Daniel BuchananSource: American Journal of Mathematics, Vol. 39, No. 1 (Jan., 1917), pp. 41-58Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370442 .

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Oscillations near an Isosceles- Triangle Solution of the Problem of Three Bodies as the Finite

Masses Become Unequal. By DANIEL BUCHANAN.

? 1. Introduction.

In an article entitled "Oscillations near one of the Isosceles-Triangle Solutions of the Three-Body Problem," which appeared in the July (1915) number of the Proceedings of the London Miathematical Society, the author of the present paper discussed the periodic oscillations of an infinitesimal body about a straight line drawn through the centre of gravity of two finite bodies of equal mass and perpendicular to the plane of their motion, which was assumed to be circular. In the problem now under consideration the finite bodies are assunmed to be of unequal mass and the third body is assumed to be infinitesimnal. The finite bodies are started so that they move in circles, and the infinitesimal body oscillates about the straight line through the centre of mass of the finite bodies and perpendicular to the plane of their motion, as in the formner article. Initial conditions are determined so that the oscillations shall be periodic. The solutions for the motion of the infinitesimal body are expansible as power series in a certain parameter E which represents half the difference in mass between the finite bodies. When E=O the solutions reduce to the simplest case of the isosceles-triangle solutions, viz., that in which the finite bodies are of equal mass and move in circles and the third body is infinitesimal.

As we shall have frequent occasion to refer to the former paper we shall refer to it as Proc., followed by the number of the equations or the section.

? 2. The Differential Equations.

Let ml and m2 denote the two finite bodies of mass mn-e and n-+E, respectively, and let yu denote the infinitesimal body. Let the unit of mass be so clhosen that m=1/2, the linear unit so that the distance from m1 to in, shall be unity, and the unit of time so that the Gaussian constant shall also be unity. Let the

6



42 BUCHANAN: Oscillations near an Isosceles-Triaqile Solution of the

origin of coordinates be taken at the centre of mass, the plane of motion of rnl and m2 as the an-plane, and let the coordinates of in1, n2 and yu be denoted

by 0l, I1, I i; 02 n 2 2 and i, n, ;, respectively. If the masses m1 and 2 are

started from the points 1/2+E, 0, 0, and -1/2+E, 0, 0, respectively, so that they will move in circles, then

0i= (i+E) COS (t-to), {2- (-E) COS (t-t0), (1) ni =(i+e) sin (t to) I72=-(8-0) sin (t-to) J

and the differential equations of motion for the infinitesimal body are

~~~~l ~~~~~~~~2~C 2 l [P 2] 2 3 p1 p3 ?[p pS R

p=-[ (E -1)2+ (n -2i2?<2],

=2 [ ( p2)2+ ( 2 piP2)2+<2]2

If we refer the motion of the system to a set of rectangular coordinates rotating about the ;-axis with the uniformi velocity unity, that is, if we trans- form the coordinates of yt by the substitutions

=x COS (t-to)-y Sin (t-to),

n=x Sin (t-to) +y CoS (t-to), (3)

then, after (1) and (3) are substituted in (2), the differenatial equations become

-2y-=-[3 33+4E-]r r3]

Z _3 [r +32 + [r3 33 |

=1 [C(2 +? -C)2+y2+Z2]r r2 - (4)

2~~~~~~~~~~~r These equations admit the J'acobian integral

(Xt)2+ (,Vl)l+ (ZI)2=Xl+y2+ 1-2E + 1+ 2E + conist. (5)



Problem of Three Bodies as the Finite Masses Become Uneaual. 43

When E=O and the infinitesimal is projected along the z-axis, then x -y_O and the differential equations (4) reduce to

(18z ( +4 Z2) y

The periodic solution of this equation is, Proc. (4),

z =q a sina (er-r) + 136 a' [sin 3 ( - -r) -sin (er-r,) + ....

? (t-+) 9 2 32 a+2 +.

The constant a is a variable parameter and a/Vl4 (1 + denotes the initial projection of ,u from the origin when e=0. The numerical values of to and xr can both be taken to be zero without loss of generality. As series similar in form to 4 occur frequently in the sequel, we shall call them triply odd series inasmuch as they contain only odd powers of a and odd functions of odd multiples of x.

? 3. The Equations of Variation.

We wish to show the existence of periodic solutions of (4) which are expansible as power series in E and which for e 0 reduce to x = y 0., z- =. Let us substitute in (4)

(t-to)= (x r-TO) Vi(1+3) z-' +u, (6)

where w vanishes with e. If we denote derivatives with respect to r by dots, and expand the right members as power series in E, x, y and w, then the differential equations (4) become

i=O i=O j=lI Y - 2 (I1 (+ 3)- A(1 + a) _ = X IW ~)+Y -s X Y()t 7 i=o i=o

j=i yj Y~O+y ~~Y~ 7 w+W0w~~W~~0~+~ ~ iooO i0 fool

W+ WoW =-2 w ( o) + XW'i Ei. io2 i=O j=f

Before characterizing the undefined terms in (7), we shall define a triply even series, as such series occur frequently in the sequel. A triply even series is a power series in a2 with sums of cosines of even multiples of fr in the coefficients. The hiighest miultiple of r in the coefficient of a2k is 2 k. Such a series is called triply even since it is even in a and only even functions of even niultiples of 'r enter the coefficients.



44 BUCHANAN: Oscillations near an Isosceles-Trian-ale Solutiont of the

The ?), Y) and WV) are homogeneous polynomials of degree i in x, and w, but only even powers of x, and y enter while w rnay enter to any power. The coefficients in XMO) and YPO) are triply even or triply odd series, according as i is even or odd, respectively. The coefficients in TMO) are triply odd or triply even series, according as i is even or odd, respectively. As far as the computations have been carried out we have, Proc. (5),

Wo = I-a' 92-9 cos 2 17 + a' ( 7-48 cos 2 7 + 87cos4s 4 +

I) = 2-3 a'(1-4 cos 2 - o) c

Y( ~~3 y00) -1 32a2(1+2 cos 2e) +.....

X(?)- 48(1+8) Aw, Y(?) =12(1+3) iw,

X(?) X2 [16-24 a' (7-10 cos 2 ) + ...]-y2[24-36 a2(2-5 cos 2r) +....] -w2[24-108 a2(4-5 cos 2e)+....],

y (0) =-x2 [24-36 a2(2-5 cos 2'r) +. ]+y2[6-3 a2(1_10 cos 2r) +....]

+w2 [6-9a2(7-10 cos 2r) +....], W(?) - -X2 [24 a sin r+... . + y2 [6 a sin r+... ] + w2 [18 a sinr+ ... .

The X Y)i) and WO') for j>0 are likewise polynomials of degree i in x, y and w, but y enters only to even degrees. If j is odd, the X() are even in x, and YO and WP are odd in x. If j is even, the X(i) are odd in x, and

YP and MD are even in x. The coefficients in X() and YW are triply even or triply odd series, according as w enters to even or odd degrees, respectively, but in WV') the coefficients are triply odd or triply even series, according as w enters to even or odd degrees, respectively. It is observed that X(i) = 0 if j is even, and Y(j =R> = 0 if j is odd.

The expansions in the right members in (7) are valid only in a certain region of convergence. It is seen from the values of r1 and r2 in equations (4) that x, y and z must satisfy the inequalities

-1< +e+e2 2e_ +X2+y2+Z2<<

-4< _e+e2?+x-2EX+2+y2+Z2< .

The region of convergence is obtained by replacing the inequality signs with signs of equality. When referred to the fixed h-axes the region is found to be the spindle formed by rotating the common portion of two intersecting circles about their common chord. These circles have their centres at 1/2+ E, 0, 0

and -1/2+E, 0, 0 and have radii 1/V2.



Problem of Three Bodies as the Finite Masses Become Unequlal. 45

The equations of variation are obtained by putting E =O in (7) and taking only the linear terms in x, y and w in the right members. They are

~- 2,\/IJ(1+8)}yj-[4(l+g)+Xo)]x-O, 1 y+2V/+(1+8) t ---[J(1+3)+YO()]y0, (8) ib?I+J0w=o.

These equations are the same as Proc. (6) if we put m0O=j in the latter. The solutions of the first two equations are obtained from Proc. (8) by putting mo=i. They are

x = A1 e'v-Pt1r ul+A2, e-/-l u2+A3 e 92r U3+A4 e-P2 u4, }r( Y ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~ ~~~~ 9) Y =V.:-1 [Al e`V=1 8T, v,-A2 el-V-l #1 -V21 + A3 eg--,, v8--A4 e-#2r V4,

where Ai (i=1, 2, 3, 4) are the constantsl of integration, ui and vi (i = 1, 2, 3, 4) are periodic functions of er having the period 2 n; and

a2k (j=1, 2)

are power series in a2 with real constant coefficients which are determined by the conditions that ui and v, shall be periodic. The values of -(f3f))2 and (3O)))2 are the roots of the quadratic

64y2-48 r-119 =0, and, therefore,

The ui and vi are power series in a2 with sums of sines and cosines of even multiples of er in the coefficients, the highest multiple of r in the coefficient of

2 a 'k being 2 k. In ui and vi (i=1, 2) the coefficients of the sines are purely imaginary and the coefficients of the cosines are real. In ui and vi (i=3, 4) all the coefficients are real. Further,

v1(-1)= v2(-V-1), v3(r) =4 ),

ui (0) = 1 (i =1, 2, 3, 4).

The determinant of the solutions (9) and their first derivatives is a constant,* and at r=0 its value is

A= 10 V-119 + terms in a2. (10)

This determinant is different from zero for a = 0 and will remain different from zero for a2 sufficiently small. Hence, the solutions (9) constitute a funda- mental set.

* Moulton's "Periodic Orbits," ? 18.



46 BUCHANAN: Oscillations nea,r an Isosceles-lTriangle Solution of the

The general solution of the last equation of (9) is the same as Proc. (7), viz.,

w- A5q+AO [z+Arq],

(p = cos -r + 196 a' [cos 3, - cosr] +...

X = sin r + l96 a9 [sin 3r + 5 sin (11)

A 9 a 141 a A=-- a2+14a+.. 2 16at.,

where A6 and A6 are the constants of integration.

? 4. Existence of Symmetrical Periodic Orbits.

Let us choose as initial conditions of (7)

X(0)=al, I;(0)=0, y(O) =O, (0)=z2, = W(O)=O, 0() a,3. (12) With these initial conditions it can be shown from the differential equations (7) that x is even in 'r and y and w are odd in xr. Hence, sufficient conditions that the solutions shall be periodic with the period 2 it are

$o=y=w=O att=z. (13)

When the conditions (12) are imposed on the solutions (9) and (11), we find that A1=A2, A8=A4, A5=O, and that A1, A3 and A6 are linear functions of

a'! (i= 1, 2, 3), vanishing with at. Now let us integrate equations (7) as power series in e and at, or, which is more convenient, in E, A1, A3 and A6. In so far as the terms wlhich are linear in A1, A3 and A6 are concerned, we obtain

x( = A1 [ e P1 T it1 + e-V71 T u2] +A[e92T u3+ e-f2 T U4], 1

y:1 = V1-i A1 [e V 'T v1-e V=1P1TV2] + A3 [ eeP2Tv3 -e-P2Tv4] t4 (14) w,, = A(; + A tip .r

When the periodicity conditions (13) are imposed on these solutions, we obtain

0 _ A1[ V-i1+u1(O)] [eV(iPz0r-e-V2I1i+A3[,32+its(O)] [eg2r-e-P2 u]

+ terms in e and higher degree terms in A1, A3, A6 and e, 0 A/- v (0) Al[ e N17r - e- lPi7r] + V3(0)A3[ e#2r-e-2 7r] (15)

+ higher degree terims in A1, A3, A6 and E,

0 = A6A Xt + higher degree terms in A1, A3, A6 and e.

The determinant of the linear terms in A1, A3 and A6 is

D = [eVi1Pi7r-e- V=iP7r] [eP22-re-2%7r]

{V3(0)[V 131+U1(0)I V Vi(0)[132+83(0)1}. (16)



Problem of Three Bodies as the Finite Masses Become Unequal. 47

The last factor has the form 5\ V-119 +

+ terms in a', 7 V2

and is different from zero for a2 sufficiently small. The second factor can not vanish as /2 is real. Then the determinant D can vanish only when g3 is an integer. When g is not an integer, D is not zero; and hence equations (15) can be solved uniquely for A1, A3 and A6 as power series in e which vanish with E and converge for 1 Ej sufficiently small. Therefore symmetrical periodic solutions of (7) exist when g, is not an integer, and they have the form

00 00 coo

x=ZXj1, y =yieE, w WjE'7 (17) j=1 j=1 2=1

where each x$, y, and w1 is separately periodic with the period 2 x in x. From the following considerations of the differential equations (7) we

show that x and y are odd in e, and w is even in e. The right member of the first equation in (7) is odd in x and E, considered together, and even in y. The right member of the second equation is even in x and E, considered together, and also even in y. Let the solutions (17) be denoted by

X=;(E), y =y(E), W=W(E).

If the signs of x, y and e be changed, the differential equations remain un- changed, and therefore,

X=-X(-e), y=-y(-e), w=w(-E)

are also solutions. Since the solutions as power series in e are uniq-Le, then

X (E) =-X ( .-E), y(E)=z-y(-E), W(E)=W(-E),

or, x and y are odd in E and w is even in E. Hence, the periodic solutions of (7) have the form

00 00 00 c = X2j+1 e2i+l, y=X Y2?1 e+, w X u'1 E2j. (18)

3=0 j=0 j=l

The same result would be obtained in the construction of the solutions if the forms (17) were used instead of (18).

? 5. Existentee of Symmetrical Periodic Orbits when g, is anz Inzteger.

When g, is an integer, the determinant (16) vanishes and the terms in (15) of higher degree in E, A1, A3 and A6 must be considered in order to establish the existence of symmetrical periodic orbits. We take the same initial conditions (12), and as in the previous section we integrate (7) as power series in A1, A3, A6 and E. In addition to the linear terms already obtained in (14), we need the quadratic terms in A1, A3 and A6, and th e termn in A, in so far as it enters x. If these terms are denoted by xc11, Yli, wl1 and cx30



48 BUCHANAN: Oscillationts near an Isosceles-l'riangle Solution of the

respectively, then all these terms except wl can be obtained from Proc. ? 6 by putting m0=1/2, X=0 in the solutions there denoted by the same notation. Thus,

Xii Al A6 Qler [ eV=19-l r uj-e V1#1 r U2

+ A3A6Q2,r[eP2TU3r-e-2'r u4], __ ~~A

Ylli= AjA6Qis[eV-PTvi+e-Vr1I31Tv2] (19)

+ A3A6 Q2r[ eP2 T V3 + e-P2T V4],

Al= Q3 [ er[Vl

tr, u1t6e-V- e T 12],

where Q,, Q2 and Q3 are power series in a2 with real constant coefficients. The term wl is the solution of the differential equation

iwil + wl- Wil = il, 1(20) where the right member is a linear function of 20, yl, W20 multiplied by a triply odd series. The complementary function of (20) is the same as the solution of the last equation of (8), and on using the method of the variation of parameters we obtain

Wii='r [A2p1+ A23] +A2[non-periodic terms] +periodic terms,

where q1 and qp are power series similar to p in (11). When the periodicity conditions (13) are imposed on the solutions

X=X10+X;1l+X30+ **

we obtain the equations

0 =A3B[92+i13(0) ] eP27r-e-P2 + V-inAoQ2[e92r+ eOr 27A

2LeQ. eJ

? A A1 -1 g3+j(0[) ][A6Q+Al Q3] + terms ine

and cubic and hiigher degree terms A1, A3 and A6,

O V=-vV3(0) A3 {eP2 7r-e-92 7r + A6 Q [eP2g7r + e-P27r]1 (21)

?vjin\ Ai{42vl0 AlQ?vA0lA3Q2[eP2?e-P2A1]}2

+ terms in e and cubic and higher degree terms in A1, A3 and A6,

0 _ n[A A6+Al 1(n) +A3q3(,n) + terms in e and higher degree terms in A1, A3 and A6.




Where the double sign occurs the + is to be taken if g, is an even integer, and the - if it is an odd integer.

The last equation of (21) can be solved for A, as a power series in A1, A8 and E which vanishes with A1 and A6 but not with e. We shall refer to this solution as (21c). When it is substituted in the second equation of (21) we obtain an equation in A1, A3 and E which vanishes with A1 and A3 but not with E, the terms of lowest degree being Al and A. As the coefficient of A3 in this equation is a power series in a2 with additional terms in 1/a2, it will be different from zero, in general, and the equation can be solved for A3 as a power series in A3 and E which vanishes with A1 but not with E. Denote this series for A3 by (21b). After substituting (21c) and (21b) in the first equation of (21), we obtain an equation (21a) in A1 and e which vanishes with these terms and in which the lowest power of A1 alone is A . The coefficient of A' in this equation is a power series in a2 with additional terms in 1/a2 and will, in general, be different from zero. Hence, (21a) can be solved for A1 as a power series in F1 which vanishes with E and converges for IEI sufficiently small. There are three solutions for A1, but only one is real, the other two being complex. When this power series for A1 is substituted in (21b) and (21c), A3 and A6 are likewise power series in E. Therefore, when 31 is an integer, periodic solutions exist having the form

X- = $() 3, y (j)3 y , W ()3 (22) j=1 j=1 j=1

where each x(i) y) and w(l) is separately periodic with the period 27t in er. These solutions converge for E sufficiently small numerically. There is only one set of real solutions, the other two sets being complex.

In the practical construction of the solutions it can be shown that x and y are odd in el, and that w is even in E&. This property of the solutions can be deduced directly from the differential equations (7) by an argument similar to that used at the end of ? 4 if we replace e in that section by E. Then we may consider the solutions to have the form

oo 2j+1 oo 2j+1 X 2j $ (2i+l)8 FE y(2j+l) E 8 W W(21E3 . (23)

j=O j=0 j=l

? 6. Proof that all the Periodic Orbits are Sytmmetrical.

We shall now consider the existence of periodic orbits in which the infinitesimal body is projected from the xy-plane in a direction which may not, be perpendicular to the x-axis, and from a point which may not lie on the x-axis. These orbits, if they exist, are called the general orbits. They have the initial values

x(O)=a-1, -a (O)=a-2, y(O)=ae3, Y(O)=a4, ?W(O)=0, '(O)=a5 * (24)



50 BUCHANAN: Oscillations near an Isosc6les-Triangle Solutiou of the

The initial value of w can be taken to be zero without loss of geometrie generality since the infinitesimal body must cross the xy-plane if the motion is to be periodic, and hence the initial time can be chosen as the time when the infinitesimal body crosses the xy-plane. Sufficient conditions that solutions having the initial values (24) shall be periodic are

x(2n)-x(0)=0, y(2rt)-y(0)=0, w(2 n)-tv(0)=0, l 25 x(27c)-x(0)=0, y(2n) -y (0)=0, iv(2 n)-t(0)=0. J

We shall now show that one of these conditions, viz., iv (2n) -w(0) 0, can be suppressed.

So far no use has been made of the integral (5). It is by means of this integral that we show that one of the conditions in (25) is redundant. When (5) is transformed by the substitutions (6), it takes the form

.2 +?2+(4++i)2+F(x, y2, w; e) =, (26)

where F is a power series in x, y2, w and e having periodic coefficients. Let us make in (26) the usual substitutions

x=x(0)+x, y=y(O0)+.y, w=0+w,

X=$(0)+$, Y= (O)+Y, tW=t(0)+w, (2)

where -x,x, y, y, w and w vanish at r= 0. We shall denote the equation resulting from substituting (27) in (26) by (26a). By putting r= 0 we obtain from (26a) an equation (26b) connecting the terms of (26a) which are independent of ,., W. On substituting (26b) in (26a) we obtain an equation

G(,xi y, y,I w,I w)- O, (28) in which at er= 0 or 27t there are no terms independent of the arguments indicated. The coefficient of W(2 n) in this equation is 2 [a+iv(0)], and it vanishes only when wb (0) =-a. If W (0) = --a then z (0) = 0, since z =, + w

and 4 (0) =a, and as z(0) =0 it follows that z-0. As we desire solutions which are not identically zero we must take w' (0) = -a. Therefore the coefficient of w (27t) in (28) is not zero at r=27t, and by the theory of implicit functions this equation can be solved uniquely for w (2 n) as a power series

W(2n) =HIx(2rn), x(2 n), y(27t), y(2t), Tw(2n7t) (29)

which vanishes with the arguments indicated and converges for sufficiently smiall values of the moduli of these arguments. When the first five conditions of (25) are satisfied it follows from (29) that the arguments of H are all zero, and therefore W (2 r) = 0. Then, since w ( 2n) = i (2 7t) - v (0), it follows that the condition tb(2n) -w(0) =0 is a consequence of the other conditions in (25) and may be suppressed.



Problem of Three Bodies as the Finite Masses Becomne Unequal. 51

We shall now integrate equations (7) as power series in ao (i= 1,., 5) and e. The differential equations from which the linear terms are obtained are the same as the equations of variation, and consequently the solutions are the sanme as (9) and (11). When the initial conditions (24) are imposed on these solutions, we find that A5=0 and that the remaining Ai are linear functions of ac, which vanish with at Then, as in ? 4, it is more convenient to integrate (7) as power series in Ai and e. On imposing the necessary conditions of (25) upon (9) and (11) we obtain

0 = Al[e2V:-lPI7-1] +A2[e-2V/=1P1w-1] +A8[e2#2-1] ]

+A4[e-2P,2 1] +.

0 = [A [e2V=-lPi7r_jj-A2 [e-25 V1 ]

+ 92 VS(O+ V83(O ) ] {As e2 #2 71]-A]-A4[e-#271-jj

0 = (3v0()) Al[e2V=lPi7r_I]-A2[e-2V-l -1] 3 +V8(0) {As[e2P2r-1] --A4[e 2 -2 1]}+

0 [=-livi(0) +V\/-1 i(0) ]{A1[e2V-IP1ff_1] +A2[e-2V-=l Pi-1]}

+ [92V2(0) +?3(0) ] {A8[e2#2r--1] +A4[e221 +

0 = 2nAA6B(0)+

The determinant of the coefficients of Ai is

2 tA A ?(0) [e2V=-#j7r_j] [e-2V=-l#i7r1] [e2#2 7r__ 1 e-2#2 7r_1] (31)

and it vanishes only when g, is an integer since 32 is real and A, A and (0) are different from zero for a not zero but sufficiently small numerically. If g31 is not an integer the determinant (31) is different from zero and equations (30) can be solved uniquely for Ai as power series in E which vanish with E and converge for IEI sufficiently small. Hence, when gi is not an integer, the general orbits exist uniquely and have the same form as the symmetrical orbits. Since the general orbits include the symmnetrical and both classes of orbits are unique, then all the periodic orbits are symmetrical when g, is not an integer.

When B1 is an integer the coefficients of A1 and A, in the first four equations of (30) vanish and in order to discuss the existence of general orbits, it is necessary to consider the terms of higher degree in (30). In addition to the linear terms (9) we require, as in ? 5, the quadratic terms in

Ai (i 1, 2, 3, 4, 6) which contain er as a factor, and the terms of x containing t as a factor and of the third degree in A1 and A2. These terms are found in the same way as the corresponding terms were determined in ?5. They have the form



52 BUCHANAN: Oscillations near an Isosceles-Triangle Solution of the

$11 A, Q, x [A, e'v:-' .f1 r- it,-A, e-':-' #17

T

%]

+ A A6 Q2 t [A3 e232T n.3-A4 e -3T ri.]

-1

Yii= A, Q, T [A 1 e 7--l pl1r v, + A2 e-: 01' ptV2 ] Yn x/|1AQ[IV-~P+AV-~P] (32)

+ -A( Q er[A e#2TvS+A4 e-[2T v4],

WI, l [A1A2q1+A3,A4 q)3] +A2 [non-periodic terms] + periodic terms,

x80 =Al A2 Q8 r [Al eV:-, 91r ul_A2 eV:l:' 1r u,].

When the necessary periodicity conditions of (25) are imposed on equations (32), (9) and (11), we obtain

2 n O =A3[e227r_j] +A4[e-2P27-1]+ + A6Q,[Al-A21

+ 2VlIAeQ2[A3e2P2r-A4e-2927r]+ 7t

A1A2QS[Al-A2]

+ terms in E and higher degree terms in Ai,

? = [92+itS(O) I {A$[e2P27r-j] -AJe-2027r-1]

+ 27t [V1ll+? (O)] [A6Q1+AlA2Q3] [A1+A2] A

+ 2\/i 7tAnQ2[92 +X (0)] [Ae2.27r+A4e-2#i r]

? terms in E and higher degree terms in Ai, o = V3(O) ASe2 22 1]-A [e-2.27r-1]}

+ 2 v1(O)A6Ql[Al+A2] (33)

+ A v3(0) A6Q2 [A3 e 2P2wr +A4 e-2P21]

+ terms in e and higher degree terms in Ai, 0= [32Vs(O )+fi3(0)]{A3[e2P2-l1]+A4[e-2P27r-1]}

+ 26/i QnA6Ql [ -l1gvl(0) +l(O)] [A1-A2]

+ A A6 Q2 [g2VS3 (0) +i'(O)] [As e2P27r-A4 e-2I%

+ terms in E and higher degree terms in Ai, 0 = 27AA60(0) +2t [AlA2cpl(0) +A8A4q3(0)]

+ terms in e and higher degree terms in Ai.




The last equation of (33) can be solved for A6 as a power series in A1, A2, A3, A4 and E, and the terms of lowest degree in A, are A1 A2 and AS A44. The deter- minanat of the coefficients of A3 and A4 in the third and fourth equations is

2 V., (O ) [d32 V3 (0) +V~3 (0) ]ILe,2#2-1] [e 2P27-1],

and it is different from zero for a2 sufficiently small since f2 is real. Hence these two equations, when A6 has been eliminated by the last equation, can be solved for AS and A4 as power series in A1, A2 and E in which the terms of lowest degree irn A1 and A2 are Al A2 and A1 A2 . When these solutions for A3, A4 and Ad are substituted in the second equation, the terms independent of E contain A A2 or A1 A2 as a factor. Hence this equation can be solved for A2 as a power series in A1 and E, and the term of lowest degree in A1 is linear in A1. Finally, when the solutions for AX, A8, A4 A6 are substituted in the first equation, we obtain an equation in A1 and e in which the terms of lowest degree in A1 is Al. As in (25a) the coefficient of As in this equation is a power series in a2 with additional terms in 1/a2 and will, in general, be different from zero; therefore this equation can be solved for Al as a power series in E* which vanishes with E and converges for 161 sufficiently small. There are three solutions as in the symmetrical orbits, but only one solution is real, the other two being complex. Since the real solutions for the general orbits and the symmetrical orbits are both unique and since the general include the symmetrical orbits, all the periodic orbits are symmetrical when 1 is an integer. Consequently all the periodic orbits are symmetrical whether 1 is an integer or not.

? 7. Constructions of the Solutions when 1 is not an Integer.

Let us substitute (18) in (7) and equate the coefficients of the various powers of &. We obtain a series of differential equations which can be integrated step by step and the constants of integration arising at eacll step can be determined, as we shall show, so that the solutions shall be periodic and shall satisfy the symmetrical initial conditions

x=y=w=O at 'r=O.

When these initial conditions are imposed on the solutions (18) we obtain

X21+1(?)-Y2j+1(0) 0, (j=O, ..o), 14 w2j(O) =O, (j= 1, o).

The differential equations for the terms in e are

xl-2\/* (1+6) ',-CA(1+3) +X(?)]x,=X('),




where X(') is the same even power series as that represented in (7) by the same notation. The complementary functions of (35) are the same as (9), and the particular integrals can be obtained by the method of the variation of parameters., The complete solutions are thus found to be

i= Af)e V-l91rui + A (l)eV-iu-1u2 +A S)eP27 U, + A (')e-#27 U4 + Cl(), 1 (35) Yi =/ V-i[A(')eV' 1#1rv,-A ()e-VJflrv2] +A ()eP2Tv. A()e -2rv4 + S (r),J

where AP')(i-l, 2, 3, 4) are the constants of integration, and C1(r) and S1(r) are triply even and triply odd series, respectively. As such series occur frequently in the construction, we shiall denote themn by Cj (r) and S (er),

respectively. Since g is not an integer in this case and 32 is real, the terms of the complementary function must be suppressed in (35) by choosing A = -0, (i=1, 2, 3, 4), and therefore the desired solutions become

X1 C1 (X), Y1 =-S1Si)r) (36-)

The differential equation for the terms in 2 iS

W2 + WOW2=W2,

where W2 is a triply odd series. The complete solution of this equation is obtained by employing the method of the variation of parameters, and it is foulnd to be

IW2 B +B2[X+Ar]ap2] +S2QC), (37)

where B(2) and B(2) are constants of integration, and P2 is a power series in a2 with real constant coefficients. In order to satisfy the periodicity and the initial conditions (34), the constants of integration must have the values

B 2) 0, Bg(2) ap2 (38)

in which case the solution (37) becomes 1- (39)

where S2(r) is a triply odd power series. The remaining steps of the integration can be carried on in the same way.

The solutions for x21.1 and Y21-1 are obtained from the particular integrals alone since g2 is real and 1 is assumed to be a number which is not an integer. These particular integrals are triply even and triply odd series, respectively, except for a factor a-2(j-2' which is introduced through the factor a-2 in (39). The general solution for the w, is similar to (37) and it can be made periodic



Problem of Three Bodies as the Finite Masses Become Uneaual. 55

by a proper choice of the constants of integration as in (38). At the steps 2j-1 and 2j the solutions are

X2j-I a2j4 C2,j1 ('),

Y2j_1= 2_ S2,-1 (r). (40)

W2j 82i 2 (f) .

? 8. Construction of the Solutions when /31 is an Inzteger.

Let us substitute equations (23) in (7) and equate the coefficients of the various powers of el. As in the previous section we obtain a series of differential equations which can be integrated step by step, and the constants of integration can be determined, as we shall show, so that the solutions slhall be periodic and shall satisfy the symmetrical initial conditions

4(0) =y(O) =W(0) O0.

When these conditions are imposed upon (23) we obtain

0(2j+D (0) Y(2j+l)(0) = (i-=O .. . . ) (41) w(2j) (0) 0, (j=1. .. ) r

The differential equations for the terms in El are the same as the first two equations of (8) if we use the superscript 1 on x and y. Since 01 is assumed to be an integer, the solutions of these equations which are periodic and whiclh satisfy (41) are

S( -A(l)~ [e\/ 1?Tui+ eVP1TU2] l (42)

y(l) =V 11TA' EeV 1pTvi eV48-1Tv2], J

where A(') is a constant which is undetermined at this step. The solutions (42) are real if A1( is real.

The differential equation for the terms in E1 iS

w') 2' + WFOW 2) =W (2)= (A (I) ) 28(2) (# (43 ) ~~ (A('~~)2S~21(r) (43

As functions similar to S(2) () occur frequently in the construction of the remaining solutions, we shall define the function S2i(r) to be a homogeneous polynomial of degree 2i in the exponentials eV/=1o1T and e-VJ1IfT of which the coefficients are power series in odd powers of a with sums of sines and \/1= times cosines of odd multiples of x in the coefficients. The higlhest multiple of 'r in the coefficient of a2k+1 is 2k+1. The coefficients of [e\IP1,T]i [e-Vf'lT ]k and




[ e V'I-fl r I k [ e - V 151, j + k 2i, differ only in the sign of \/-1. The part of

SM' (r) which is independent of the exponentials is a triply odd power series. If the exponentials are expressed in trigonometric form, then

S(2i) (er) = j

a2j+lS(21in 2/1(2?fr S'2t'(t)=5E 1 Ea hlS 2i ;sin 12ho, +7(21 +1) I e, j=0 1=0 h=O

where S(2i), are real constants: that is, S'2i' ('r) is the same as a triply odd power series except that the highest multiple of r in the coefficient of a21+1 is not 2j+1 but 2(j+ ifl)+1.

The general solution of (43) is found in precisely the same way as (37) was obtained. It has the form

w'(2' B(2) + B 2) [ X+Aerfp-a (A ()) 2 (2'),T,+ (A(') )2S (2)(t (44)

where p(2) is a power series in a2 with real constant coefficients and S(2) (xr) is similar to S(2) (er). On imposing the periodicity and the initial conditions (41), we have

B (2) = 0, g(2) _a (A'(I) ) 2p (2).45

Then the desired solution of (43) takes the form

w (2) = 12 (A(1)) 2S(2) t 1

where S32' (r) is similar to S'2) (r). Thus A(') remains undetermined at this step also, and the terms in 3 mnust be considered.

The left members of the differential equations for the terms in e2 are the same as the first two equations in (8) if we use tlhe superscript 3 on x and y. Let us denote the right members by X(3' and Y(S), respectively. Then

X(3)=(A(')) 3 (3) + X() Y(3) - (A (1)) 3a (3) . (46)

The function X(l) is the same triply even power series as that denoted in (7) by the sanme notation. The functions p(2i+l) and G(2i+1) are homogeneous polynomials of degree 2i+1 in the exponentials e,-Vltr and eVW1lpiT, and the coefficients are power series similar in form to ul and U,2 in (9). In p(2i+1) the coefficients of [e'V=1,T] [e- V=1,T]i and [e8-'] ic[eV-1T] j j+k= 2i+1, differ only in the sign of V-i; while in a(2i+l) the coefficients of [e V=12T] i [-e1-2 1l]k

and [eV=1jT]k[_e-V=1jT]i differ only in the sign of V-1. The complementary functions of the differential equations in x(? and y'3)

are the same as (9). Since X'8' and Y(8) contain terms which have exactly the same period as the periodic parts of the complementary functions, the par-




ticular integrals will contain non-periodic terms. By the method of the variation of parameters we find that the general solitions for x:3 and y13) are

-;(3) A(3)eV1=1Tul + A<3>)e-8'-1'02U+ A) efP2Tu3 + A 6)e-2Tu4 1

+xr [ (At()

) 3P(3) + P'3) j [e Vl=.rt,u

- e 7-1 3,1]

+ pW

(8 --1 [A (3)e ' - vl -A(3) + A 3)e2Tv3 -A (3)e62Tv4

+ V-Pr[ (AI))3P(3) +P23)] [eeVvI f+ e-VP1Tv2] + V-\_ ,(3j

whiere AV> (i=1, 2, 3, 4) are the constants of integration, PP3) (i= 1 2) are power series in a2 with constant coefficients, and ) and a() are similar to 3

and aW , respectively. In order that the solutions shall be periodic, then

A (3)'- -A(3)-0, p(8) (A (1)) 3+ P(3) =0.

The solutions for A(') are A(')=P(1), (,)p-(1) or ('2p(l)

where P(l) is a power series in a2 with real constant coefficients, and , c2 are the imaginary cube roots of unity. As the imaginary solutions for A(1) lead to imaginary orbits, we retain only the real solution. Then, on imposing the initial conditions (41) on (47) we find that A3()-A(3),and the desired solutions for x"( and y' become

= (3)=A(-) [e V Io, 7u + e- -l3ITu2 ] + P()

s 3 1A(3) [ee-1nlTv1- e- 1i1Tv2]+V A

The constant A(3) remains arbitrary at this step and the terms up to Ji must be considered before it can be determined. The solutions for x(' and y'3' are real provided A(3) is real, which is found to be tbe case.

The general solutions of the differential equation in Ed is obtained in the same way as (44) was found, and it can be made to satisfy tlle periodicity and initial conditions by a chioice of the constants of integration as in (45). When these conditions have been imposed, the solution for W(4) takes the form

w(4) [A()(2)+ S'4']

The differential equations in es are the same as (8) in the left members if we use the superscripts 5. If we denote the right members by X`5 and Y(5),

respectively, then

k-'6' A [A(3)p(3) +P(5)], y(&) V=-i [ A '3)0) +C55)] a2 ~~~~~a2

where p(3) p(5) and (3) (5) are similar to p(2k+1) and a(2k+1), respectively. The 8



58 BUCHANAN: Oscillations near an Isosceles-Triangle Solution of the, etc.

complete solutions of the equations in x$ and y(6) are readily found in the same way as (47) were obtained. They are:

- (6) A (5) e 'V-1'nlul + A 6) e-113ju2 + A 2)'e,rTu + A e-pTu4 1

+ -fi [p(s)X(3)2+ [eAV1fl Tui-e-V A -lTu2] + '4

(5 V- l A5e/'sA5e'-pT2 + A )SM3Ae2s (48 )

+ er [P(5)A(3) + p_(5) ]e_\/-llrU1 + e r

a 2 2 [e14v+e4T 2] + p

whlere p(6) (i-i, 2) are power series in a2 with real constant coefficients. Thle solutions for x) and y(6v can be made periodic by putting A(54=A8 O, and then choosing A(3) so that _() 3 ()o

from which it follows that A(3) is real. F1rom the initial conditions (41) we have A(6)-A(5)e

There M aining stepofe integration can be carried on in the same way. The solution at the step 2j can be made to satisfy the periodicity and the initial conditions by a proper choice of the conlstants of integration arising at that step. The solution for W21 contains the constant Af2'-1' whlich is not determined until thle step 2j+1. 'The non-periodic parts of the complete solutions at the step 2j+1 are similar to those in (48), and A 2p-' enters these solutions with the same coefficient as A(3) enters (48) except for thle factor 1/a2i-2. The solutions can be made to satisfy the initial and periodicity conditions by putting AA(2j+=A A (21+1) 4(2 1L' (2j+1L.

1 2~~~~~~

and then determinie the Ap2sf so that the coefficient of cr shall be zero. The desired solutions at the steps 2j and 2j+1 are

(2jmn) _i t A 2j+1_ [ T__e n-eriodic parts o

ya=e2j+1a -1Ar tt+1)[e iln ve48PlalraA] 2-1 + t hso

Thus the integration can be carried on to any desired degree of accuracy. The solutions of (7), whether as power series in E or id, must satisfy the

integral (5) identically in ? or E. Thus this integral, beside+s being of use in proving that all the periodic orbits are symmetrical, serves as a check on the computations.

QUEIEN'S UNIVTERSITY, KINGSTON1 O)NTARIO, N ovember 15, 1915.



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