Revista Brasileira de Física, Vol. 10, NP 1, 1980
Lagrangian Solutions to the Three-Body Problem with
Forces r - P (p integer)+
R. DE AZEREDO C AMP O S +
-
and P. LEAL FERREIRA
Instituto de Física Tedrica. São Paulo
Recebido em 18 de Agosto de 1979
The exact s o l u t i o n s t o the three-body problem i n C e l e s t i a l Mechanics, due t o Lagrange ( t r i a n g u l a r s o l u t i o n s ) and Euler (col l inear
so-
l u t ions) , are general ized t o the case of forces rP ( p b e i ng an i n t e g e r )
The s t a b i l i t y o f the system i s a l s o i n v e s t i g a t e d i n a l o c a l sense
.
(small
v a r i a t i o n s about steady motion) f o r t r i a n g u l a r and col l i n e a r s o l u t i o n s a n d
c o n d i t i o n s r e s t r i c t i n g the values o f p f o r which t h e r e a r e s t a b l e o s c i l l a t o r y modes o f v i b r a t i o n are obtained. Furthermore,
for
under consideration, Bohr o r Bohr-Sommerfeld q u a n t i z a t i o n
the
solutioris
is performed
and compared, f o r some cases o f i n t e r e s t , w i t h the WKB approximation, der i v e d from an Hamiltonian o f the system obtained by reducing i t t o a one-body problern under the a c t i o n of a c e n t r a l f o r c e a t the
o f mass
system's c e n t e r
.
As soluções exatas do problema de t r ê s corpos na Mecânica Ce-
l e s t i a l , devidas a Lagrange (soluções t r i a n g u l a r e s ) e Euler (soluções co-
L ineares) , são general izadas para o caso de forças r-P ( p i n t e i r o )
. A es-
t a b i l idade do sistema é também investigada no s e n t i d o l o c a l (pequenas variações em t o r n o da solução) para soluções t r i a n g u l a r e s e col ineares,
e
são o b t i d a s condições sobre os valores de p para que correspondam a modos
de vibração o s c i l a t ó r i o s e s t ã v e i s . Além
disso, para as soluções em
tão, as quantizações de Bohr e de Bohr-Sommerfeld são efetuadas e
radas, para alguns casos de interesse,
+
++
aproximação WKB, deduzida
Work done wi t h p a r t i a l support by FINEP (Contract 522/CT)
With a f e l l o w s h i p o f FAPESP, São Paulo.
.
ques-
compade um
hamiltoniano do sistema o b t i d o reduzindo-o a um problema de 1 corpo sob a
ação de uma f o r ç a c e n t r a l no c e n t r o de massa do sistema.
l t i s well-known t h a t the three-body problem i n C e l e s t i a l Me-
chanics cannot be solved,
i n general,
i n a ctosed form.
The d i f f e r e n t i a l
equations o f motio,; of the system make the problem o f the 1 8 order.This
~ ~
tti
system of e q u a t i m s can he redcced t o the 8
o r d e r by rneans o f the i a t e g r a l s corresponding t o c o n r e r v a t i o n of the c e n t e r o f mas5, energy and ang u l a r momenturn. ~ a ~ r a n ~pr-oved
e'
that i t i s possible t o
blem t o the
reduce
the pro-
o r d e r and no f u r t h e r r e d u c t i o n has been ohtained.
Lagrange a l s u showed' t h a t ,
i f che t r i a n g l e
fornied
by
the
three c e l e s t i a l bodies always remains s i m i l a r t o i t s e l f , then the three-body admits exact plane2 s o l u t i o n s and the bodies describe c o n i c
sec-
t i o n s , sim i l ã r t o each ather, w i t h the c e n t e r o f nas5 being a t one focus.
I f the bod,ies are c o l l i n e a r ,
t h e r e a r e a l s o exact s o l u t i o n s ,
which were f i r s c derived by ~ u l e r " ~ .
We r e c a i 1 t h a t these c i a s s ica: s o i u t i o n s , ciiie t o Lagrarige aad
~ u l e r ~
have
,
astronomical s l g ~ iicance:
f
a) t h e system
composed
by
the
sun, J u p i t e r and a T r o j a n a s t e r c i d forms an e q u i l n t e r a l r r i a n g l e and wa5
found t o be a r e a l i z a t i o r i o f the Lagraiigian s o l u t i o n ; b) t h e " g e g e n s ~ t i e i n " , t h a t i s , the patch o f 1 i g h t (a1 ready observed
astronomersj due t o
e a r t h and t h e
SUP
d
conceritration o f meteors on
by
the 1 i n e
laçt
century
through the
(c01 1 inear s o l u t i o n )
[ n the present note, i t i s shown i n s e c t i o n 2,
t h a t Laqrange
and E u i e r s o l u t i o n s can be generaljzed t o the case o f f o r c e s p r o p o r t i o n a l
t o r?
( p i ntecjer!
.
I n view o f the above examples o f r e a l i z a t i o n o f those
tions i n nature ( f o r
SO~LI-
newtcnicliz forces i .e p = Z), It i s r e a s o n a b l s t o
Z 2, these s o l u t i o n s may have a physical
suppose t h a t a i s o i n the case p
s i g n i f i c a n c e , even I n the microscopical domain. This c o n s i d e r a r i o n m o t i -
v a t e d us t o s t u d y t h e s t a b i l i t y ( s e c t i o n 3) and t h e q u a n t i z a t i o n ( s e t t i o n
4) o f those s o l u t i o n s , S e c t i o n 5 i s devoted t o some f i n a l remarks.
2. GENERALIZED LAGRANGE AND EULER SOLUTIONS
We assume t h a t t h e two-body f o r c e s between t h e bodies j and k
a r e c e n t r a l and p r o p o r t i o n a l . t o r-p
jk '
so t h a t t h e t o t a l f o r c e a c t i n g on
t h e body k i s g i v e n by
For p = 2 and C
d i e d by Lagrange and E u l e r .
jk
= G rn
ri
j 'k'
we have t h e c l a s s i c a l case s t u -
F i r s t , we n o t e t h a t i t i s easy t o extend t o any v a l u e
of
p,
p # 3, Carathéodory% p r o o f 3 t h a t t h e p l a n e formed by t h e bodies i s f i xed i n s p ace
g
.
For each t y p e o f s o l u t i o n ( t r i a n g u l a r o r r o l l i n e a r ) we
shall
c o n s i d e r two cases:
a) Cfr.m~Zar( o r "steady riotion") : t h e mutuãl d i s t a n r e s between t h e bodies
a r e c o n s t a n t s and t h e bodies r o t a t e about t h e i r c e n t e r o f rnass
with
an
a n g u l a r v e l o c i t y w.
b) no%-d2scuZm; ( o r ltpe,nZod<cll)
: t h e r a t i o s between mutuai d i s t a n c e s r e main c o n s t a n t and t h e o r b i t s o f t h e b o d i e s v a r y a c c o r d i n g t o
the
value
of p.
2.1. Lagrange iype sa9ufions
a) C i r c u l o case
The e q u a t i o n s o f motion, r e f e r r e d t o t h e c e n t e r o f
g i v e n , i n C a r t e s i a n c o o r d i n a t e s by
rnass
are
I f (Ck,qk)
(xkSYk)c o o r d i n a t e s
rotates w i t h angular v e l o c i t y w w i t h respect t o the
and
if z
=
Sk + < o k , t h e n
i t f o l l o w s from
(2.1.1)
that
D e f i n i n g now
we g e t , f rom (2.1.2),
that
The l a s t equat i o n expresses t h a t t h e c e n t e r o f mass i s
are consistent i f
(2.1.3)
where
M
i s t h e t o t a l rnass M =
It follows that, f o r
ml + m, + m,.
p Z - 1,
1
at
rest
.
Eqs
.
For p = -1, t h a t i s , f o r h a m i c forces we have
and C
jk
=
u2 K1m n a s o l u t i o n e x i s t s f o r a r b i t r a r y r
3 k'
jk'
a n d m . = m o r if C
= G m m
3
jk
j k'
and t h e bodies form an
one sees t h a t r 1 2= r23 =
Wenotice that, i f C
t h e n frorn (2.1.5),
ik
= C
equilateral triangle.
b) ~ o n - c i r n * bcase
L e t us suppose C = C and m. = m (equal rnasses case).
Let
jk
3
t h e i n i t i a l mutual d i s t a n c e s
(x.,y.)
be t h e c o o r d i n a t e s a t t = to,r
3 3
jk
and ( 5
.) t h e c o o r d i n a t e s a t t ( w i t h mutual d i s t a n c e s prik,p
being t h e
3 3
p r o p o r t i o n a l i t y f a c t o r ) . We can w r i t e
.,<
53. =
nj
3
cose
= p(x. sino
3
-
y . sino)
3
y . cose)
3
where 8 i s t h e r o t a t i o n a n g l e .
Si nce
we a r e lead, from (2.1.7)
and (2.1.8),
Equations (2.1.9)
to
a r e necessary condi t i o n s f o r t h e e x i s t e n c e
o f s o l u t i o n s . They a r e c o n s i s t e n t i f
and
~ ~ u a t i o n s ( 2 .11)
.1
are verified if
9%= a
And f r o m (2.1
.9) and
and
( 2 . 1 .11),
3
a=-
,p+I .
we g e t :
The bodies, a t any i n s t a n t , f o r m an e q u i l a t e r a l t r i a n g l e . The
r ã t i o s between mutual d i s r a n c e s remain equal and c o n s t a n t ( f o r
r e f .8)
.
2.2. Euler type solutions
A s i m i l a r t r e a t m e n t l e a d us t o t h e e q u a t i o n s
Let us define
p -2, se2
Fig.1
-
Three-body configuration corresponding to Euler solution w i t h equal
niasses, in the circular case.
We t h e n o b t a i n f r o m (2.2.1),
under t h e assurnption t h a t C
N o t i c e t h a t p = 1 v e r i f i e s (2.2.3)
jk
f o r any p
= C and
m.= m
3
arid corresponds t o
= C and m . = m, t h e r e a r e s o l u t i o n s f o r any p ,
3
c o r r e s p o n d i n g t o t h e s i t u a t i o n o f F i g . 1 . I f p = -1, (2.2.3)
becomes an
Thus, i f Cjk
i d e n t i t y and so one has a s o l u t i o n f o r a r b i t r a r y ~ . .
.i
As i n s e c t i o n 2.1 b ( n o n - c i r c u l a r Lagrange s o l u t i o n s ) , wecan
derive
I n t h i s case, one must have
and (2.2.4)
i s verified i f
Furthermore, we have
2
I f p = - 1 , a = 3 a-(p+l),
which c o i n c i d e s w i t h
the value o f
a f o r t h e t r i a n g u l a r case (2.1 .12).
I n t h e n o n - c i r c u l a r case ( t r i a n g u l a r o r c o l l i n e a r ) i t i s p o s sible, f o r p = - 5 ,
-3, - 1 , 0, 2, 3, 4, 5 a n d 7, t o s o l v e ( b y c i r c u l a r o r
e l l i p t i c functions) the i n t e g r a l t h a t gives the o r b i t s o f the b o d i e s l O .
I f p=2, t h e o r b i t s a r e c o n i c s w i t h t h e c e n t e r o f mass a t one
focus. F o r
p = -1, t h e o r b i t s a r e s t i l l c o n i c s b u t t h e c e n t e r i s a t
the
c e n t e r o f mass.
3. STABILITY OF THE SOLUTIONS
3.1. Stability of Lagrange type solutions
For C
jk
steady motion,
= G rn
j
rr
we s h a l l c o n s i d e r small v i b r a t i o n s
'k'
about
i n t h e c i r c u l a r case.
L e t us t a k e t h e o r i g i n a t m 3 and, as c o o r d i n a t e s ,
tances r 1 3 ,r Z 3
and t h e angles
r i a b l e s we have: rlz,
appear e x p l i c i t l y .
el, 8,
al
and a, as i n F i g . 2 .
the
dis-
As a u x i l i a r y va-
and 0 3 . The i g n o r a b l e c o o r d i n a t e does
not
Fig.2
- Coordinate system and auxiliary
variablasfor Lagrange's steady mo-
tion.
W r i t i n g the equations of motion i n terms o f those
coordina-
res and t a k i n g s o l u t i o n s near Lagrange's steaciy motion, we can d e r i v e
g
a
secular determinant f rom which r e s u l t s the condi t i o n (pf3) :
where K stands f o r the f r e q u e n c y ' o f o s c i l l a t i o n and M i s the t o t a l mass.
The r o o t s o f (3.1.1)
are
(a) X2 = 0, which corresponds t o the a d d i t i o n a l ignorable coordinate.
(b) K2 =
-(3-p)
(C) K2 = -(3-p)
,
f o r which there are s t a b l e o s c i l l a t o r y rnodes i f
M
7
which corresponds t o s t a b l e modes i f
For the case p=3, we v e r i f y t h a t there a r e no s t a b l e rnodes.
67
Thus,
(3.1.2)
and (3.1.3)
are the conditions o f s t a b i l i t y f o r
Lagrange t y p e c i r c u l a r s o l u t i o n s .
3.2. Stability of Euler type solutions
L e t us suppose C
a, and a,,
jk
=
r n 2 ~and t a k e t h e
coordinates
r13,
-M 2 3 ,
as i n Lagrangeis case. The E u l e r t y p e s o l u t i o n s correspond t o
the values
r 1 3
= a,
As i n s u b s e c t i o n 3.1,
~2~
=
2a
,
al = a,
= wt
w e l o o k f o r s o l u t i o n s near
motion. From t h e e q u a t i o n s o f m o t i o n , we o b t a i n ,
.
the
steady
t o the f i r s t order,four
e q u a t i o n s i n t h e s m a l l v a r i a b l e s , which l e a d t o a s e c u l a r d e t e r m i n a n t
3
whose r o o t s a r e g i v e n by
where
Thus,
t h e c o n d i t i o n s f o r s t a b l e modes a r e
f rom b) :
p < 3
;
f rom c) :
We conclude t h a t o n l y i f p < -3, t h e r e i s s t a b i l i t y f o r E u i e r
type c i r c u l a r solutions,
i n t h e equal masses case.
4. SEMI-CLASSICAL QUANTIZATION QF THE SOLUTIONS
I n t h i s s e c t i o n , we p e r f o r m
the
q u a n t i z a t i o n o f t h e genera-
l i z e d s o l u t i o n s , a p p l y i n g f i r s t Bohr q u a n t i z a t i o n t o t h e s i m p l e s t c a s e o f
c i r c u l a r s o l u t i o n s and Eohr-Sommerfeld q u a n t i z a t i o n f o r t h e n o n - c i r c u l a r
ones, f o r b o t h Lagrangian and E u l e r i a n t y p e cases.
4.1. Bohr's quantization
L e t us suppose C j k = C and
vens i n t h i s case, f o r p
# 1,
m = m. Thc? t o t a l energy E i s g i j
by
The a n g u l a r marnentum w . r . t
t h e c e n t e r o f mass i s
I t f o l l o w s , f c r p f l and pZ3, t h a t
If
p = 3 , we have E = O . F o r p = l , c o r r e s p o n d i n g t o
r i t h m i c two-body p o t e n t i a l
the
loga-
b) EuZer t y p e circular solutions
F o r p#1 and ~ $ 3 ,we o b t a i n i n a s i m i l a r way,
express i o n f o r t h e quant i z e d energ i e s (equa 1 masses case)
the
following
.
Again, i f p=3, we have E=O and f o r p = l , we a r e l e a d t o
and (4.1.6) , t h a t f o r p=2 one ç e t s a B a l -
We see, f rom (4.1.4)
mer-1 i k e spectrum. N o t i c e a l s o t h a t ,
and (4.1.6)
i f p=-1 (harmonic f o r c e s ) ,
(4.1 .4)
g i v e r i s e t o i d e n t i c a l expressions.
4.2. Bohr-Sommerfeld quantization of non-circular solutions
An i m p o r t a n t f e a t u r e o f t h e n o n - c i r c u l a r s o l u t i o n s i s t h j t t h e
c o r r e s p o n d i n g e q u a t i o n s o f motiori can be d e r i v e d from
an
Harc i 1 t o n i a n
where t h e c a n o n i c a l v a r i a b l e i s q = p ( p i s t h e a l r e a d y d e f i P
ned r a t i o of t h e mutual d i s t a n c e s between t h e bodies) and t b e c o n j u g a t e
H(qp,pp),
6.
momentum p = ma2
Indeed, f r o m t h e e x p r e s s i o n s f o r t h e k i ~ e t i cand poP
t e n t i a l e n e r g i e s o f t h e system, we o b t a i n
) I +$ '
C1
=
t
(I
( t r i a n g u l a r case)
+
2 ~ ) a - ( ~ + (c01
" 1i near case) .
T h e r e f o r e , f o r those s o l u t i o n s ,
t h e system o f t h r e e b o d i e s i s
e q u i v a l e n t t o one body under t h e a c t i o n o f a c e n t r a l f o r c e a t t h e system's
c e n t e r o f mass.
I n o r d e r t o accomplish t h e Bohr-Sommerfeld
quantization,
=
p d , where p i s o b t a i n e d by
P
J P P
P
t h r o u g h t h e equat i o n ~ ( p ) = E. The a c t i o n i n t e g r a l can
we i n t r o d u c e t h e a c t i o n v a r i a b l e J
means o f (4.2.1)
then
' P ~
be s o l v e d by t h e m e t h o d o f r e s i d u e s , f o r t h e c a s e s p = 2 , p = - l
p = 3.
a) Lagrange t y p e s o Z u t i c n s
We quote t h e f o l l o w i n g r e s u l t s :
i ) p = 2,
a r e s u l t which c o i n c i d e s w i t h t h a t o b t a i n e d f r o m (4.1.4)
ii)
f o r p = 2;
for p = - 1,
Again,
t h e r e s u l t c o i n c i d e s w i t h (4.1.4)
b) E d e r t y p e
f o r t h i s v a l u e o f p.
soZutions
We
I ) f o r p = 2,
i i ) f o r p = -1,
o b t a i n the f o l l o w i n g results:
and
4.3. The WKB approximation
S t a r t i n g from the Hami l t o n i a n ( 4 . 2 . 1 ) ,
which describes theso-
l u t i o n s under consideration, we can w r i t e the corresponding S c h r t j d i n g e r
qquation, whose eigenvalues a r e the quantized energies o f the system. For
instance, f o r t h e Newtonian case ( p = 2 ) , i n the unequal masses case,
one
can r e a d i l y derive, from well-known quantum-mechanical r e s u l t s ,
ex-
the
press ion:
v a l i d f o r the Lagrangian s o l u t i o n .
The same formula can be obtained be means ofBohr o r Bohr-Sommerfeld quantization conditions.
We may apply the WKB approximation t o get energy
i n some s i g n i f i c a n t cases ( f o r i nstance, the l o g a r i thmic
p=l , and the 1inear p o t e n t i a l case p=0)
expressions
p o t e n t i a1
case
.
With the Hamiltonian (4.1.1),
we have
where
I n the WKB approximation f o r the three- dimensional problem,we
have
1, 2,
Considering the case o f S-waves
Zutions are:
...
(4.3.4)
the r e s u l t s f o r Lagrrmgs type so-
For
n
= 1,
(4.3.6)
gives
( n o t i c e t h a t t h e e x a c t f a c t o r i s 2.3381).
E u l e r t y p e sohtions, we o b t a i n :
For the
2
i) p = 1
,
Veff = 3C l n
ii) p = O
,
Veff = 2 CP ;
P
po
'
The above r e s u l t s a r e t o be compared w i t h those o b t a i n e d i n sect i o n (4.1)
f o r p = O and p = 1.
5. FINAL REMARKS
We have shown t h a t i t i s p o s s i b l e t o g e n e r a l i z e
Lagrange
and
E u l e r t y p e e x a c t s o l u t i o n s t o t h e three- body problem f o r h r c e s p r o p o r t i o n a l
t o r-P ( p i n t e g e r )
.
The c o n d i t i o n s o f s t a b i l i t y o f t h e c i r c u l a r s o l u t i o n s have a l s o
been e s t a b l i s h e d f o r a r b i t r a r y values o f
p.
For
p
= - 1 (harmonic f o r c e s ) ,
near s o l u t i o n s f o r a r b i t r a r y r
jk'
c01 1i -
t h e r e a r e t r i a n g u l a r and
T h i s o c c u r s because, as i s w e l l known,the
probiem i s always s e p a r a b l e . E q u i l a t e r a l t r i a n g u l a r s o l u t i o n s
are
s t a b l e , even i f t h e masses a r e d i f f e r e n t . C o l l i n e a r s o l u t i o n s
masses a r e n o t s t a b l e . The q u a n t i z e d e n e r g i e s (equal masses)
i n Lagrange s and E u l e r ' s case ( c i r c u l a r and n o n - c i r c u l a r ,
aiiways
w i t h equal
are identical
see
4.2.3
eqs.
and 4 . 2 . 5 ) .
For Newtonian f o r c e s ( p = 2 ) wi t h equal masses,
not stable
see eqs. 3.1.3
and 3.2.2).
the solutions are
I n b o t h cases, a Balmer-1 l k e f o r m u l a
i s ' o b t a i n e d , eve6 i f t h e masses a r e d i s t i n c t (see f o r instante, eq.
We n o t i c e t h a t
i t was
shown
g
that
particlilar
4 .?. 1) .
solutions also
e x i s t f o r t h e case o f f o u r - b o d i e s w i t h f o r c e s p r o p o r t i o n a l t o r v , w h i c h
are
analogous t o t h e Lagrange and E u l e r s o ! u t i o n s h e r e discussed f o r t h e 3-body
problem.
F i n a l l y , we t h i n k i t would be i n t e r e s t i n g t o s p e c u l a t e about t h e
possible existence o f microscopical s y s t e m r e a l i z ing the
E u l e r i a n s o l u t i o n s i n t h a t domain.
Lagrangian o r
However, no a t tempt i n t h i s d i r e c t i o n
was made h e r e .
REFERENCES AND NOTES
1
. Lagrange,
J .L.
"Essai s u r l e Problême des T r o i s Corps" (17721, Oeuvres,
v o l . V I , Paris.
2 . A p r o o f t h a t t h e movement i n t h i s case o c c u r s i n
a fixed
plane
g i v e n by Lagrange, and i n a more elementary f o r w by Carathéodory
was
(see
ref. 3).
3 . Carathéodory, C . "Uber d i e Strengen Losungen des Dreikorperproblems",
S i t . Bayr. Akad. Wiss. 257, 1933.
4 . E u l e r , L . "De motu r e c t i l i n e o t r i u m corporum se mutuo a t t r a h e n t i u m " ,
Novi Comm. Acad. S c i . Imp. P e t r o p . 11, 144 (1767).
5 . S i e g e l , C.L.:
lin,
Vorlesungen Über l-limmelsmechanik, Springer- Verlag, B e r -
13.56.
6 . Some a u t h o r s r e f e r t o b o t h o f them as Lagrange s o l u t i o n s .
Following
H. P o l l a r d ( r e f .7),
we p r e f e r t o d i s t i n g u i s h between E u l e r ' s ( c o l l i n e a r )
and Lagrange's ( t r i a n g u l a r ) case.
7.
Pol l a r d H.:
Mathematical I n t r o d u c t i o n t o C e l e s t i a l Mechanics,
t i c e - H a l l Inc.,
8. Moulton, F.R.
New Jersey,
Pren-
1966.
: C e l e s t i a l Mechanics, M a c m i l l a n Comp.,
London,
1947.
9. R. de Azeredo Campos, " Soluções Lagrangianas do Problema de 3 Corpos
com Forças p r o p o r c i o n a i s a
Tese de Mestrado
,,I FT,
10. W h i t t a k e r , E.T. : h a l y t i c a l Dynamics, Dover Publ.,
São Paulo, 1979.
New York, 1944.