ON
LAGRANGE
SOLUTIONS THREE
V. T.
RIGID
IN
THE
PROBLEM
OF
BODIES
KONDURAR
Civil Engineering Institute, Dniepropetrovsk, U.S.S.R.
(Received 16 December, 1973) Abstract. The present paper is a direct continuation of the paper (Duboshin, 1973) in which was proved the existence of one kind of Lagrange (triangle) and Euler (rectilinear) solutions of the general problem of the motion of three finite rigid bodies assuming different laws of interaction between the elementary particles of the rigid bodies. In particular, Duboshin found that the general problem of three rigid bodies permits such solutions in which the centres of mass of the bodies always form an equilateral triangle or always remain on one straight line, and each body possesses an axial symmetry and a symmetry with respect to the plane of the centres of mass and rotates uniformly around its axis orthogonal to this plane. The conditions for the existence of such solutions have also been found. The results in Duboshin's paper have greatly interested the author of the present paper. In another paper (Kondurar and Shinkarik, 1972) considering a more special problem, when two of the three bodies are spheres, either homogeneous or possessing a spherically symmetric distribution of the densities or of the material points, and the third is an axially symmetrical body possessing equatorial symmetry, the present author obtained analogous solutions of the 'float' type describing the motion of the indicated dynamico-symmetrical body in assuming its passive gravitation. In the present paper new Lagrange solutions of the considered general problems of three rigid bodies of 'level' type are found when the axes of geometrical and mechanical symmetry of all three bodies always lie in the triangle plane, and the bodies themselves rotate inertially around the symmetry axis, independently of the parameters of the orbital motion of the centres of mass as in the ' float' case. The study of particular solutions of the general problem of the translatory-rotary motion of three rigid bodies, which are a generalization of Lagrange solutions, is in the author's opinion, a novelty of some interest for both theoretical and practical divisions of celestial mechanics. For example, in recent times the problem of the libration points of the Earth-Moon system has acquired new interest and value. A possible application which should be mentioned is that to the orbits of artificial satellites near the triangular libration points to serve as observation stations with the aim of specifying the physical parameters in the Earth-Moon system (e.g., the relation of the Earth's mass to the Moon's mass for investigating the orientation of the satellite, solar radiation, etc.).
1. The Force Function of the Problem Let us consider a system of three rigid bodies Mo, M1, 3//2 possessing arbitrary constant masses mo, m l , m2, an arbitrary constant external form and a definite internal structure, consisting of elementary particles which are attracted according to the Newtonian law. Let us select the bodies M~ and Mj (i, j = 0 , 1, 2; i C j ) and assume that they have no common part. L e t f b e the gravitational constant and Pi and Pj any points of the bodies Mi and Mj respectively in which the elementary attractive masses are concentrated. Let Celestial Mechanics 10 (1974) 327-344. All Rights Reserved Copyright 9 1974 by D. Reidel Publishing Company, Dordrecht-Holland
328
V. T. KONDURAR I
dmt = 6i dzt,
dms = fis drs
where c~t and 3s are the spatial, in general variable densities of the bodies M~ and M s at the points Pt and Ps, and dzt and dr s are the differential elements of volume. Then the attraction force function Uts of the two bodies M~ and Ms, mutually attracting each other according to the Newtonian law, is determined by the following proper integral taken over both masses mt and rns of the bodies Mt and M s (Kellog, 1929) Uts = f f
~ dmt dins
(1)
M z MS
where A~j= Ajr is the distance between the current points Pt and Pj of the bodies Mi and M s. The magnitude of the integral (1) for the given form of the bodies and the given density distribution inside the bodies depends upon the position and orientation of one body with respect to the other, or in other words, upon the position and orientation of each of the two bodies with respect to a certain system of coordinate axes Ogr/( having a fixed direction and with origin at an arbitrarily chosen fixed point in space. Let us construct an analytical expression for the force function (1) transferred to this system of absolute axes O~r/( which is not connected with either of the bodies M~ or Mj. Let at, bt, c~ and as, bs, c s be the coordinates of the points Pt and Ps of the bodies Mt and M s. Then in terms of these coordinates, we have A,~ = (aj - at) 2 + (bj - bt) 2 + (cj - ct) 2
(2)
Now let us introduce for each body Mt a proper system of coordinates GiXt YiZt with origin at the centre of mass Gt and with axes directed along the principal axes of inertia. Then the absolute coordinates at, bt, ct of the point P~ of the body Mt will be related to the coordinates Xt, Yt, Zt of the same point in the proper system by means of the transformation (i)~ a(i)i~ _(t)=~ at = ~i + a11..~i + ~2) i + u13,-i ,,,fl i).~ _(DIS bi - ~ + -2,~t + t/22.)'i -t-
U23-t,
_( i) ~
,,1(t) V ~(t)i5 Ci - - ~i 71- t~31"~i -~- t232.,vi ~
aaa,-t,
(3)
(i)~
where ~t, rh, (t are the absolute coordinates of the inertia centre G, of the body Mr, and a~ts)(k, s = 1, 2, 3) are the direction cosines of the proper axes of the body Mt with
329
O N L A G R A N G E S O L U T I O N S I N THE P R O B L E M OF T H R E E R I G I D BODIES
(i) respect to the absolute axes. These cosines aks are connected with Euler's angles by the well known formulas
a(/) 11
- - COS g t / C O S ( 0 i
a(i) 12
=
- c o s g q sin(& - sinv~ cos(o~ cosO/,
=
sin ~ / s i n
__
sinv~ cos(o~ + cos q// sin r cosO/,
al'/) 13 a(i) 21
a(i) 22 =
-sin
- sin gh sin(o/cosO~,
0/,
sin (oi + cos ~ t : cos (oi cosO/ ,
~r
(4)
a(/) 23 = - c o s ~v/sin Oi, a(O 31 = sin (O/sin 0/ a(O 32 = c o s ( o / s i n O i , a(/) 33
--" C O S 0 i ,
where V/ is the precession angle, 0i the nutation angle and (O/ the angle of proper rotation. In view of formulas (3) and (4) for the body Mi and similar formulas for the body Mj, the force function (1) is a well known function of the six coordinates of the centres of mass G / a n d Gj and the six Euler angles of the bodies M / a n d Mj (5)
~/, r/i, 4/, q/i, 0i, (O/; ~j, r/j, 4j, ~uj, 0j, (Oj.
Now, if we consider the system of coordinates with origin at the centre of mass of one of the three bodies and assume that its axes are respectively parallel with the axes of the fixed system O~r/(, then the force function (1) becomes a function of nine parameters. Indeed, let X1, Y1, Z1 and Xz, Y2, Z 2 be the coordinates of the centres of mass G, and G2 of the bodies M1 and M2 in the system GoXYZ with origin in the centre of mass Go of the body Mo and with axes parallel to those of the absolute system O~r/~. Then we can write x, = c j / - ~o,
y/ = r / / - r/o,
z, = ?,'/- ~ro.
(6)
Note that ~0
- - O,
Yo=0,
Zo-O
and, therefore ~i --- X j
~
Xi
r/j
7]i
. a
(j - (/ = z i - z/.
yi,
(6')
Thus the expression (2) can be presented as z
=
Fij
--- X j
(7)
7i5 +
where X i _~ ,n(J)-~
~ll"~J
/v(J)u
_3f_ ~ 1 2 Y j
/v(J)u
"J- t * 1 3 z ' j
--
(i)--
allxi
--
(i)--
al2yi
_(t)-- tllaZi
(8)
330
v . T . KONDURAR
and the corresponding expressions for Yu, Zu are obtained from (8) by cyclic permutation of the first subscripts of the direction cosines and of the coordinates x, y, z. From this it follows that the integral (1) is a function of the nine independent variables Xj
-
-
Xi , yj
-
-
y~ , zj
0
-
z~ ;
~u~, 0~, ~0~, ~ui, 0j, ~j.
(9)
The Forces of Mutual Attraction and Their Moments
Let us consider that M~ and Mj do not collide, have finite sizes and continuous densities. Then the force function U u determined by the integral (1) and its partial derivatives of any of the arguments (5) or (9) are finite and continuous. They determine the projection of the forces acting on the body Mi(Mj) due to the body Mj(Mi) and their moments with respect to the centres of mass G~(Gj) along the coordinate axes and along the axes of precession, nutation and proper rotation. When differentiating the function Uu with respect to its arguments, we may apply the rule of differentiating a multiple integral with respect to a parameter. Let us take the variables (9')
Xi, Yi, Zi; ~i,Oi, (,Oi
as parameters and calculate the derivatives of U u by the variables (9'), considering the function U u as a complicated function depending on these variables through the intermediate variable A u. Then by differentiating the quantity A/} ~ introduced under the integral sign in (1) on the parameters X~, Y~, Z~ we find
ozl;j'
Z,
OXi
A 3ij
L, Oyi
OAij c~z~
-- i
A 3ij
2i j
A.3lj
Likewise, by differentiating A u-1 for example on (Pi according to (7) and (8) we obtain ~Au- 1 ~(0i
_A-3(~ij
OXij
O Yij
~,ZLj.]
where ~Yij
-- xi
,-, (i) Oall
~f,(i) ~'12 t- >-~
-
~_(i)
6 r
l- z i
and by analogy for the derivatives on ~i, 0i, the unwritten expressions for the derivatives of Yu and Zu may be obtained from the last expression for the derivative of )~ij by a cyclic permutation of the first subscript of the direction cosines. From formula (4) we have
331
ON LAGRANGE SOLUTIONS IN THE PROBLEM OF THREE RIGID BODIES
~,,(~) =
"
(3u/ ~
--
als
(i)
c;a3s
(i)
t"Ct2s
,
O,
--
c~u/ i
3aki (~)
"(~) sin~oi ,,(ofsin~ui (k ; = - 3 ~ \ _ cos ~, (30i = - ~
1) = 2),
9
,,,(~)fsin ~ti (k ~k 1) = a ~ ) cos~o, = ,,32),_cosgti = 2), 3al.~ (9
,, (i) ua23 ~
--- ~_(i) 3 3 s i n ~bri ,
(3argO)_ a (k!Z) (3qh '
_(i) C O S ~ r i ,
~/'/33
(30i
,,d a k(,) 2. &Pi
._.
,, (i) ua33
,,_,(i)
-,.kl,
,.t~ak3( i ) (3q)i
(30i
--
--
sin
0 i ,
-- O.
We shall also have similar expressions for the derivatives of the quantities uks _(i) on the variables ~uj, 0~, ~oi. Substituting the last expressions in the former equalities, we find
(3Au- 1
A13
_
(3g' i
[,.,(i)~
kUll~'i
-~-
a(i)= (i)~. " 1 2 Y i "-~ a 1 3 ~ - i ) Y i j
(i)(a(2i~xi + a22Yi
--
,,.,(i)=,
n t-
"~
t*23/-'il/xij,
-1
(3AiJ
(i) (i)(i)~ A 3t j . c~0~ = (a31x~ + a3zyi + a33z~)(Xu s i n ~ i - Yu cos~) +
+
A .a t2.
(3Au- 1
x,t~33('t(i)s i n a i
= (a~ff~-
"xi
"Jr" (./33_(i)COSfpi.jTi
a~Yi)J~u + i""~(i)\-22.,~i ~
(i)-
--
_ sinOi.z,i)Zij
a21yi)(i)- Yu~ +
r,(i)~ ~
+ (a3zX~ -- ,-,31.rij~u.
N o w taking into consideration the known properties of the direction cosines (4) and the indicated differentiation rule, we find the following formulas oV, j
(3xi
- Xu
~Uu =
ff
Y,j
(3Yi
~Uu (3zi
I F~udm'dmj
M, Mj (Zij =
Zij (10)
oE ~U~j ~i(i
~Uij where
--
fMf fM
dmidmj Aij
'
332
v . T . KONDURAR
~olj
"-- ( Y j -
-
- - Y i -4- a21.,,~ j nt-
(Xj
X i -[-
-
-'~ " 1 2 . Y J -~
.ll.~j
sin
+
a,2yi
+
,-,~a,-u
"13~j3\"21"~i "~
Oi.Y~i)+
+ [xj-
x~ + ,-.(~)~ . ~ j + "~(~)~ ..~2~j + a(,~)~) sin ~/i -
-
Yi +
(Yj -
/I(1) -~ ~,z~;
+
,o(J)i'; .q_ /v(J)~r "l ~,22~j ,.:a,.~ cosr
x
(11)
x ( a a , x i + a 3 2 Y l + aaazi) ~ij
=
--
,.,(0 (j) "" (i) cos~o~yi a335zj)(a3a sin~oi'2i +-33
,-~(J) -~ ,-,(J) 77 zi + - a , - ~ j + ,-32.rj +
-.,i j = (zj -
-
~23.-,jlU,~ll.a,i
(Xj
Xi +
9.v(J) -~
~11"~j
~(J)~
Av " 1 2 Y j
"~
+ (Yj -
Yi + -, (2J1)-~~ + •22fj .~(i)~
+
z, +
(zj
-
9,v(J)~.
+
,,v(J)T;
" a(l~)Szs)(a(l~2i
a(l~)Yi) +
+ a(2~)zj)(a~Xi " "
-
+
"
"
_
(i)a21yi) + aalYi)
.
Both integrals (10) and integrals (1) extended over the body masses M~ and Mj are functions of the quantities (9). Obviously, the quantities (10) determine the projection of the resultant of all the attractive forces acting on the body M~ applied in the centre of mass Gi and its moment with respect to point G~, along the axes X, Y, Z, and along the axes of precession, nutation and proper rotation. Obviously the derivatives of U u along Xi, Yi, Zi are equal, according to (6), to the derivatives with respect to ~i, rh, (i
,.
cqG;
cqGj
Zu
~Uu *
/7(J) we find By changing the index i to j in the formulas and consequently a~ i] to --k~ analogous components for the body Mj. In other words
OUu
0Gj
0Uu
0Uu
8x~
8xi
8~j
8~i
and similarly for the derivatives along yj, z i, 0j, ~0j. 3. Equations of M o t i o n
Denoting the projection of moments by P u , Qu, Ru applied to the body Mi on its proper axes we can write the following known formulas Pu
= (~u
Qu
= (~u
Rij
~
sinqh
-
-
cos0~-~j)
cos~oi sin 0t
cos~0~.Zu, (12) sin ~0i"=%j,
~)ij"
Note that Z u is determined by (11). * H e r e a n d in the next section, a sans-serif H used for a capital eta, a n d a sans-serif Z for a capital zeta.
333
O N L A G R A N G E SOLUTIONS I N THE PROBLEM OF THREE R I G I D BODIES
Then the equations of motion in our system of three rigid bodies in the absolute system of coordinates can be written as follows m,~, = ~,,
AiPi
mi/j, = H,,
-
(B~ -
Ci)qiri
= Pi,
Bigh -
(C~ -
A,)r,p,
=
Ciii
(Ai
Bi)piq~
= R~.
-
-
m,~, = Z,,
(13)
(14)
Q~,
The rights members of Equations (13) and (14) are determined by the formulas 2
2
Y_., = ~. 2..t 3 ,
H i -~
2
EH
Z,
ij,
j--O
j=O
j--O
jr
j-r
jr
2
2
(15)
~Z,,
--
2
P,= j =Ee,,, O
o,= j =Eo,,, O
jr
jr
R i ---
~
Rij
j=O jr
where
Here p~, q~, ri are the projections of the rate of angular motion of the body M~ on its proper axes, and A~, B~, Ci are the principal central moments of inertia of the body Mi. The ordinary Euler kinematic equations Pi =
d/i
sinqh sinOi + O~cos~0t, 9
q~ = q} cos ~0~sin 0i - 0~ sin (0i,
(16)
r~ = ~ cos(pl + ~0~
should be joined to the system (14). Equations (13), (14) and (16) form a system of the 36th order with 18 unknown functions ~ , ~h, (i; g/i, 0i, q~ (i=0, 1, 2). This system of equations in the general case of arbitrary bodies has only ten first integrals which are analogous to the classical ones (Duboshin, 1968). Therefore this system of equations, in general, is impossible to integrate completely. From the given integrals for lowering the order of the system, as in the classical three-body problem only the six integrals of the motion of the centres of mass prove to be convenient. Let us transform our system of equations to relative coordinates. Such a transformation to the axes G o x y z with origin in the centre of mass Go of the body Mo will change only Equation (13), while the other equations of the 18th order (14) transform according to (6) or alternatively to (6') and thus remain unchanged. The initial system of nine equations (13) with unknown {~, rh, (i can be reduced to a system of six equations of the 12th order with unknown functions xi, y~, z~ (i= 1, 2) and to an additional system of the 6th order with three unknown functions go, ~/o, (o which can be integrated after integrating the system of the 30th order. Indeed, according to (6) and (13) we have 1 x i =
i--
o =
1 ,= i -
mi
1 ,=o
mo
2
=
1
2
k
llqi k = 0 kr
i,.
t7"10 k = 1
1
334
v . T . KONDURAR
Now Equations (13) are transformed, as in the classical case, to the form 2i = Xi,
Yi =
El,
gi = Z1,
22 = X2,
Yz = Yz,
s = Z2.
(18)
The equations for y~ and s are written according to the obvious analogy. Here according to (17) 1
ml
(X o +
1
mo
(Xo + Xo ) (19)
x2-
1 m2
(X o +
1 mo
(Xo + Xo2)
and analogously for Y~, Z~ where X~j, Y~j, Zij are determined by the formulas (7), (8), (10), in which Xo = 0, Yo= 0, Zo= 0. Equation (18) together with Equation (14) are a system of the 30th order for determining the unknown quantities xl, y~, Zl, x2, Y2, Z2, Ni, Oi, (Pi (i=0, 1, 2) which has four integrals three of moments of the rate of motion and an integral of energy. We shall not write out these integrals, nor shall we use them to lower the order of the new system, since this would result in cumbersome equations.
4. Plane Solutions of the Problem
Though, in general, it is impossible to integrate completely the system of equations of the general problem of three rigid bodies, this system, under certain conditions, permits solutions which are analogous to the Lagrange and Euler solutions in the classical three-body problem. Our solution determines the non-Kepler orbit of each bodypoint with respect to each other body or the centre of mass of the body. The existence of some similar solutions in the general planar problem of three rigid bodies was proved for the first time in the above mentioned paper (Duboshin, 1973). The search for other particular solutions of this problem is certainly of great interest. Let us consider the problems which correspond to the motion of the centres of mass Go, G1, G2 in a fixed plane, for example, in the plane xGoy which does not cause a loss of generality. In order that the gravity forces applied to the arbitrary body Mi (i=0, 1, 2) should give rise to plane motions of its centre of mass in the plane z =0, it is necessary that the projection of the principal vector of the force on the z-axis and principal moment of the force with respect to the centre of mass on the axes of nutation and rotation be simultaneously equal to zero at Z i = Z j = 0 . Thus, in the case of the existence of plane motion, the identity Z 1 ~ 0)
Z,---0
should be fulfilled for any value of t~> to, or in the expanded form, according to (19)
335
ON LAGRANGE SOLUTIONS IN THE PROBLEM OF THREE RIGID BODIES
1 ml
(Z,o + Z,2) =
1 (Zol
mo
-]- N o 2 ) ,
(20) 1 m2
(Z o +
=
1 /no
Formula (10) for Zi~ gives at zl Zis = f
f
f ra(J)~ ~, 31"a'j
(Zo, + Zo2).
~Z2~0 ,,(J)~
,,(J)y,j
"At- "*32.Yj AV " 3 3
_
a~a~X,
_
(i)i7 _ a(3~5,) x
aa2a,
i
M~ M,I
dm~ dm~ x A3
(20')
ij
where Aij is determined by the expression (7) under the condition that & - z i - O . Let us now introduce the proper rotating coordinate axes for each body Mi by the rotation of axes formulas xi = )2~ cos~0i - y~ sin ~0i,
35~ = )2~ sin rp~ + yi cos~0~,
z~ = zi
and also, let us suppose that each body Mi possesses a dynamic-geometrical symmetry with respect to the principal plane z - 0 . Thus the external surface of the body is symmetrical with respect to this plane (xy). In addition, the density of each body is an even function of the distance of the current points (ffi, ~ , :Y,) of the body M~ to the plane z = 0. Let us consider the case in which 0o = 0 1 - 02-89 during the whole time of motion of the body M~. Then the proper axis Zi of the body M~ lies permanently in the plane xGoy (together with the new axis 2'~), and the density of the body M~ in each point is considered as a function of its coordinates (ff~, y~, ~,~), being an even function with respect to the new ordinates y~ fii = 6,(X~, . ~ z , ~ ) .
(A)
In this case cos 0i = 0 for any t >1to and the expressions (20') are simplified and reduced to the form _ y~) dmi dmj
Ab
(20")
M l Mj
where due to the formulas (7) and (8) A ~zs = (x s -
xi + :Tj. cos Vs + z~ sin ~s -
X~ cos V, - U sin Vi) 2 +
+ (Ys - Y, + :Tj. sin ~s - z~ cos g/s -
+ (y~ - y;)2
x, sin ~
+ z~ c o s ~ , ) 2 +
336
v . T . KONDURAR
Here ff~, fi~, ~ are the new coordinates of the mass element dmi =Oi dff~ d35~d ~ of the body M~ in its proper rotating axes, a~ being an even function with respect to Y~' from (A). We can see that the quantities A~j do not change with the changing of indices Y" and Yj. Therefore all the integrals (20") due to the condition (A) are equal to zero. Let us consider now the integrals (10) determining the projections of the force moments 2..,j and R~j and prove that they are also identically reduced to zero for the studied plane motions under the given conditions (A) imposed on the structure of the bodies. We introduce into the formulas (10) and (11) two systems of coordinate axes with their origins at the centres of mass G1 and G 2 of the bodies 21,/1 and M 2 and obtained from the coordinate system Goxyz for the body Mo by a parallel displacement of axes. Let us write the corresponding formulas of rotation for these axes ?
~1
--1
xj = xj cos gtj + zj sin ~tj.
y~ = xj sin ~uj - zj cos ~uj, Then the formulas (10) and (11) give
-'~'ij
--
[Yi Xj
- x ~ + xa)sin g ~ - Yl(Ya-
Yi +
Yj) X
M~ M j x
cos ~,
-
yje;]
dm~ dmj A 3 ij
(21)
R
,=syf
[ - - f i ~ ( X j -- X i "Jr- X j ) COS ~ff i - - Y i ( Y j -- Yi -at-Yj)
•
M s Mj
x sin ~, + yj.ff~]
dmi dmj /I 3
ij
Here now z,~,. = ( x j -
x, + xj' -
x',) '~ + (y~ -
y , + yj' -
y~)e + ( y j -
y,)-' ~.
(22)
These integrals contain the factors Y~ (k = i, j) behind the integral expressions, but are independent of X~ and Z~. This means that they, as well as the integrals (20"), are all equal to zero, which is needed. Thus the general equations of the translatory-rotary motion of three bodies possessing a dynamic-geometrical symmetry in the form of (A) with respect to their proper planes ( y ' = 0) permit plane motions in which the centres of mass move in a fixed plane coinciding with the planes of body symmetry and each body in general, rotates around its proper Z-axis precessing in the plane. In order to write the equations of the plane motion problem let us refer to the general formulas (10) determining the component of the principal vector of forces on the axes
ON LAGRANGE SOLUTIONS IN THE PROBLEM OF THREE RIGID BODIES
337
x and y, and composing the principal moment of the forces on the precession axis Z. These formulas give !
-
x~ + x j -
!
x~)
dm~ dmj '
M t Mj
(23)
_ Y, + y j _ y~) d m , d m j A3
ij
'
M l Mj
[x;(yj - y, + yj) - y ; ( x j -- x, + xj)]
dm~
dmj
A 3 i.l
M l Mj
The integrals (23) in distinction to (20"), (21), (22) do not contain factors Y[ behind the integral expressions and therefore under the conditions (A) do not vanish. Obviously the component moments of forces on the proper axes X~ and Y~ of the body Mi according to (12) will be given by }[ti j sin~oi,
P ij =
Q ij =
~ t i j c o s (tgi .
(24)
By introducing now the polar coordinates p~, v~ and P2,/)2 and points G~ and G 2 and by transforming Equation (18) for the case of Z1 =0, Z2 =0, we can write the whole system of equations of the plane problem in the form i 5 1 - bzpl = X~ cosy1 + Y1 sinv~,
d (p~bl) = p ~ ( - X ~ dt fi2 - -
sin/)~ + YI cosy1), (25)
/32p2 -- X'2 c o s / ) 2 21- Y2 sin v2,
d (p~.b2) = P z ( - X z sin v2 + Y2 cos v2), dt A~Pi
-
(Bi
--
Ct)qiri
-- Pi,
B,O, -
(C~ - A~)r,p~ = Q~,
C/'i
(At
-
-
Bi)piqi
=
(26)
O,
where according to (16) pt = ~, sinfez,
q~ = ~t cos ~oi,
(27)
Here due to (15) and (24) P~ = 7ti sin~0i,
Q~ = 7~ cos~o~
(28)
where 2
(29) j=O
338
v . T . KONDURAR
The quantities 7t~ under the condition (A) in general depend upon p~, P 2 , Vl, /')2, g~, ~0~ and completely determine the right members of the equations of the rotary motion of the bodies under discussion. Let us note that we have imposed the condition (A) on the form and structure of the bodies. This condition is in general a very strong one if we consider the angle ~0~to be a variable quantity. Indeed, when changing ~0i in the interval from 0 to 2n the condition (A) requires the realization of the symmetry of each body M~ with respect to any plane passing through the axis of rotation of the angle ~0~. But if the bodies possess this symmetry, then the external surface of each of them is the surface of rotation around the axis Zi, and the density has the same axial symmetry, that is, it depends arbitrarily upon the distance of any point from the axis of rotation =
+
+ y- -,' ,2 z- t, ) .
=
(B)
In this case since the choice of the direction of the principal axes in the equatorial plane is arbitrary, we may take any two planes perpendicular to each other passing through the centre of mass as these axes. Thus, any two mutually orthogonal planes passing through the axis of symmetry of each of the three bodies are its symmetry planes. On the basis of condition (B) by substituting ~i, Yi for -y~, 2~, we obtain A~-B~. Consequently, the third equation (26) gives the integral (0~=(0~~ determining the inertial rotation of the dynamic-symmetrical bodies. The first two equations (26) after their multiplication by sin~0~, cos~0~ and cos~0~-sin~0~ and addition, are reduced to the form 2
c,J/,(o, = O,
(30) j=O jg:i
where the quantities 7t~i are given by the expressions (23). Due to the formulas (22) determining A~j they do not depend upon the angle ~0~.From the first equation (30) it follows that three cases are possible. The supposition, however, that both factors are zero all the time is of no interest, since all the bodies would be at rest with respect to their centres of mass; consequently, only two cases remain (1) ~o, = O,
r o,
(2)
= o,
/o, r o.
In these cases the rotation of the bodies possessing dynamic-geometrical symmetry is possible only around their proper axes of inertia. Indeed, without loss of generality, let us suppose that in the first case ~=(p}o)=0. Then from formulas (27) we shall find that p~--0, q~= ~ , ri=0. Consequently, the second equation (30) determines the precessional motion (around the axis y~) of the body of rotation possessing an axial symmetry with respect to the axis Zi. The right members of the equation do not depend upon ~0i, but only upon the quantities Pl, P2, V1, Va and certainly upon all the angles g~ determining the position of the centres of mass G1 and Gz with respect to Go and the orientation of the axes of rotation of the bodies in the plane xGoy.
339
O N L A G R A N G E SOLUTIONS I N THE PROBLEM OF THREE R I G I D BODIES
In the second case we have the inertial rotation of the bodies around their proper axes of symmetry =
+ (t-
(31)
~
if only all the 7%=0. The latter identity for the considered case ~,i=~u~~ is self explanatory as is directly seen from the expression (23) for 7tu under the additional condition imposed upon the density of the bodies with respect to the z-axis -t2 6, = 6,(.22 + 37~ z, 22) = 6,(.2',2 + y--t2 , , z, ). (C) For proof, taking for the sake of definiteness a constant value gi for example ~ , = 0 , we have ~="2~, y j = -17;. Then, according to (C), substituting Z~ for - Z ~ and retaining X~ and Y~, we obtain 7%=0. We find the same supposing that p,,=89 I.__ X[,=Z[,, Yk X[,. This means that in the considered case each body possessing dynamic-geometrical axial symmetry rotates inertially (31) around the axis of symmetry which is displaced parallel to the plane ( X Y ) together with the non-rotating body (precession). This translatory-rotary motion of the bodies does not depend upon the motions of the centres of mass of the bodies in the plane ( X Y ) . The equations of motion of the centres of mass are determined by the formulas (25), where )(1, X2 and Y~, ]I2 are given by the expressions (19) and (23). For the considered case of the rotationally symmetrical body (C), it is obvious that the integrals containing the factors X ~ = . ~ or Y [ , = - Z [ , ( k = i , j ) behind the integral expressions are equal to zero, independent of the values ~0kif g,, = 0 or g'k--89 Therefore the expressions (23) for X, s and Iris are simplified and take the form X,s = (xs - xi)Fis,
(32)
Y,J = (Ys - Yi)Fis
where
F,s=fff dm'dm 71-g- 9
(33)
tj M~ M j
Here the mutual distances between the particles of the body A,s are determined by the formulas (22) in which it is necessary to replace the rectangular coordinates Xk, Yk by their expressions in polar coordinates Pk, Vk, and (0k by the expressions (31). Now all that remains is to calculate the right members of Equations (25). Replacing V1 and V2 by the new variables V--V1 and ~u= V2-V1 taking into consideration (19) and (32) and effecting the necessary simplification, we obtain instead of (25) the following equations
[51
--
d
fi2 m
ply 2
----
Pl\ m ~
4
ml
}
Fiq + P2 ml/
\ml
F~COS~/,
mo/
Fo2] sin ~u,
= PlPz\ ml
mo !
p (e + ,))2 =
(Fo t
-Pekmo
d dt [p2z(b + ~)] = P~P2
(34) m2
m2 /
F211 sin ~. m2]
+ Pl
F~
\ rn2
mo
!
COS ~ ,
340
v . T . KONDURAR
We notice that both Equations (34) and the original Equations (25) do not vary when substituting V for V+ ~z. One more important fact should be noticed: the integrals (33) do not change if we set ~0~=0. Therefore, the quantities Fij in the right member of Equations (34) should be considered as known functions of the independent variables Pi, P2, gt, v. In reality, let us calculate the integrals (33) in supposing that the centres of mass of the considered bodies are in the planes (XY). For this we shall consider the force function of mutual attraction (1), where Aij are determined by the formulas (22) for ~k = 0. Thus Az
-
(xj
-
x,
+
Xj. -
X',) 2 +
y~ + ~ -
(yj
z-j)2 + (Yj - y~)~
(35)
where Xo - - O,
Xi = p i C O S V i ,
X2 = p 2 C O S V 2,
(36) Yo = 0,
y~ = p~ sin v~,
Yz = P2 sin vz.
F r o m formulas (1) it is obvious that, for example auoi _ auto_
api
api
f j"
OAol]
Aox3(Aol ~
-f
] dmo dmi
Mo M1
where Aoi
0Aol 1
•Yl
OX 1
(x~ + .~', -- Xo) ap~ t- (y~ + Y'; - 5;) Op~
or according to (36) Aol
OAoi ap~
= p~ +
07~
-
Xo)cosvi + (Zo -' -'
-
z- 'i ) s i n v i .
Likewise we obtain analogous formulas for the two other pairs of indices. Hence, taking into account that the integrals
If
-' dmiA ~i.i ,
M t Mj
f f -' dm'A3
(k = ~,j)
M l Mj
are equal to zero, we find Foi =
Flo =
10Uol
--, Pi OPi
1 0Uio
Pi Opl
,
w h e r e P i , P 2 , A = %/p~ nt- p2z -
10Uo2
1
Fo2 =
--, P2 0P2
F12 -
/72o =
1 0U2o --,
F21 = -
P20pz
2pip2 cosgt are the sides of the
0U12
--, A OA
1 0U2i
"A ad
,
triangle GoGiG2.
(37)
ON LAGRANGE SOLUTIONS IN THE PROBLEM OF THREE RIGID BODIES
341
Equations (34) with the unknown functions pl, P2, v, ~ under certain conditions have the Lagrange (triangular) solution gt = + 89
P~ = Pz = p(t),
(38)
in which the centres of mass G~ and G2 of the bodies M1 and Mz describe similar orbits with respect to the centre of mass Go of the body Mo, forming together with Go an equilateral triangle with in general variable sides. The bodies themselves lying in the triangle plane and possessing an axial and equatorial symmetry rotate uniformly around the symmetry axis. The necessary conditions for the existence of such solutions, of which we can easily be convinced, have the same form as that in the paper (Duboshin, 1973) 1
Fol =
/H o
1
Fzi,
m 2
1
ml
Fio =
1 m2
F2o,
1
1 Fi2 = - - Fo2 ml mo
(39)
or are the same according to (37) and (38) 10Uol mo Op
10U21 rn2 Op 1 mi
0U12 Op
1 OUlo ml Op 1 0Uo2 mo Op
10U2o Op
(4o)
where Uij are the force functions of each pair of bodies, ml their masses, and p the triangle side. Equations (40) have a simple mechanical sense: they are the conditions which the structure of the rotational symmetry of the bodies should satisfy. Homogeneous spheres or those possessing a spherical structure are simple examples for which conditions (40) are reduced to identities. Conditions (40) for bodies of rotational symmetry, either homogeneous or with rotational symmetrical distribution of densities and with correspondingly parallel principal axes of inertia, are given by Duboshin (1973) A o - Co A1 - C1 A2 - Ca = = , mo ml m2
(41)
where A i--Bi, C, are the inertia moments of the body Mi. The preceding necessary conditions (40) for homogeneous ellipsoids of rotation with semi-axes a~, bi are also sufficient. It is obvious that in this case they are reduced to the form a 2 - c~ = a 2 - clz = a 2 - c2z.
(42)
This last result (42) has also been obtained in the paper by Vidyakin (1972), where the author, considering the motion of three homogeneous spheroids with a common symmetry plane, has studied only the translatory motion of the given b o d i e s - the motion of their centres of mass, deferring the question of rotary motion of the bodies around their centres of mass.
342
v . W . Ir
The results of Duboshin (42) have been obtained as a particular case of the general problem of translatory-rotary motion of three rigid bodies with arbitrary laws of interaction between the elementary particles of the bodies. If conditions (39) are fulfilled the points Gt and G 2 will describe around Go orbits determined by the equations p2f3 = C , -
=
[Fol
-p/--
F12
+
\too
+
ml
(43)
F~o~
.
,
mE/
where C is the arbitrary constant of integration of the surface areas. We shall notice that two Lagrange solutions, in conformity with ~ = 6 0 ~ and - - 6 0 ~ as in the classical case, correspond to any solution of the systems (43). We can still notice that among the Lagrange solutions there can be permanent solutions determining the circular orbits of the points G1 and G2 with respect to Go with radius p = a and with constant angular velocity 17= I7o determined by the formula
c
/Fo
bo = a2 = + . i ' ~ mo
t
ml
~ -- . m2
(44)
When calculating the quantity Fi~ one should bear in mind that I//i =
0,
(/)i =
0,
~
=
-t-89
/91 =
/)2 =
A = a.
Since the quantities F~ are positive, our problem has a cyclic Lagrange solution at any value of p = a , except during mutual collision of the bodies.
5. Duboshin's Case
Now let us weaken the initial condition (A), supposing that throughout the motion the quantity ~0i remains constant. For the sake of simplicity let the proper axis X~ of each body M~ coincide with the nodal line of the plane ()7~y~) on the plane ( x y ) , i.e., we shall suppose that fp~=0 for any value of t >1to. Then we shall replace (A) by the weaker condition
(A') This condition means that each body has only one symmetry plane coinciding throughout the motion with the plane ( x y ) . It is obvious that there exist bodies having three unequal principal central moments of inertia (A~r B~r that can satisfy this condition. Under the additional condition ~0~=0, the proper axes of the body M~, rotating x~, - y~ - and not rotating xl, -' yl, -' coincide because Pl = 0, qi= q)~, r~ =0, Pi =0, Q~ = ~ , R~=0.
ON LAGRANGE SOLUTIONS IN THE PROBLEM OF THREE RIGID BODIES
343
Then the first and third equations of the rotary motion (26) become identities and the second, as above, gives the equation
or in the expanded form
Bo~o = ~ol + ~o2,
ai~i = B~
(45)
~r)'lO "Jr- ~[';12,
= e~o + ~'~1.
The right members of (45) are given by the expressions (23), where x~,, ys are determined by the formulas for the angle of rotation of g/k under the condition that rpi =0. Consequently -' = 2k, xk
-' = 35k, Yk
(46)
-' = zk. zk
The equations of the translatory motion (25), i.e., the equations of motion of the centres of mass G~ and G2 in the plane (xy), are retained and together with the equations (45) form a correlated system of seven equations of the second order with seven unknowns, P l , P 2 , V l , V2, ~/0, ~r
~/2,
determining the translatory-precession motion of the three bodies. This motion possesses symmetries (in the form and distribution of the masses) with respect to the plane (xy), on which the centres of mass G1 and G2 describe orbits around Go. The bodies themselves precess around the axis perpendicular to this plane. To disclose the solution of the plane problem obtained in the paper by Duboshin (1973), we shall subject the distribution of densities of the bodies to the law
6, = a,(x'? + el ~, Y?) = ,~(x'? + y;~, yl ~)
(c')
where the quantity (46) should be taken into account. In this case the considered bodies Mi possess a geometric-dynamic symmetry with respect to the axis of inertia, perpendicular to the symmetry plane, where the centres of mass of the bodies are permanently situated. , A - ~j 3 , Aij - 3 , taken over Due to this, the multiple integrals of the expressions Xk" or yij. the bodies Mi and Mj, are equal to zero. Therefore, the .quantities J(~j and Y~j are again reduced to the form (32), and all gt~i=0. Then the Equations (45) give ~=0
Oz
~,, = ~,~o) + r
to)
i.e., each body, as in the case of (31), rotates uniformly around its axis of symmetry perpendicular to the fixed plane of the centres of mass, irrespective of their motion in the plane. The obtained particular solutions for the general problem of three rigid bodies are
344
v.T. KONDURAR
of definite interest and value for, firstly, they appear to be closely connected with the important specific properties of the general solution of the problem with the necessary number of arbitrary constants, and secondly, they are the generators for certain periodic solutions. Therefore they have direct astronomical application. References Duboshin, G. N.: 1968, Celestial Mechanics (in Russian), Nauka, 2nd ed. Duboshin, G. N.: 1973, Celes. Mech. 8, 495 (in French). Kellog, O. D.: 1929, Foundation of Potential Theory, Berlin. Kondurar, V. T. and Shinkarik, T. K." 1972, Bull. ITA, XIII, 102. Vidyakin, V. V.: 1972, Astron. Zh. 49, no. 6.
