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one of classical problems in astronomy and physics is the three - body problem in the newtonian gravity ( e.g. , @xcite ) .
the gravitational three - body problem is not integrable by the analytical method .
as particular solutions , however , euler and lagrange found a collinear solution and an equilateral triangular one , respectively .
the solutions for the restricted three - body problem , where one of three bodies is a test mass , are known as lagrange points @xmath0 and @xmath1 @xcite .
_ lagrange s equilateral triangular solution _ has also a practical importance , since it is stable for some cases .
lagrange points @xmath2 and @xmath1 for the sun - jupiter system are stable and indeed the trojan asteroids are located there . for the sun - earth system , asteroids were also found around @xmath2 by the recent observations @xcite .
recently , lagrange points have attracted renewed interests for relativistic astrophysics @xcite , where they have discussed the relativistic corrections for lagrange points @xcite , and the gravitational radiation reaction on @xmath2 and @xmath1 analytically @xcite and by numerical methods @xcite .
it is currently important to reexamine lagrange points in the framework of general relativity . as a pioneering work ,
it was pointed out by nordtvedt that the location of the triangular points is very sensitive to the ratio of the gravitational mass to the inertial one @xcite .
along this course , it might be important as a gravity experiment to discuss the three - body coupling terms in the post - newtonian ( pn ) force , because some of the terms are proportional to a product of three masses as @xmath3 .
such a triple product can appear only for relativistic three ( or more ) body systems but can not for a relativistic compact binary nor a newtonian three - body system .
it was shown by ichita et al .
, including the present authors , that a relativistic equilateral triangular solution does not satisfy the equation of motion at the first post - newtonian ( 1pn ) order except for two cases @xcite ; ( 1 ) three finite masses are equal and ( 2 ) one mass is finite and the other two are zero .
hence , it is interesting to investigate what happens at the 1pn level for three unequal finite masses in lagrange s equilibrium configuration . for the restricted three - body problem ,
on the other hand , krefetz found a relativistic triangular solution by adding the corrections to the position of the third body @xcite .
for three general finite masses , we shall look for a relativistic equilibrium solution that corresponds to lagrange s equilateral triangular one .    throughout this paper ,
we take the units of @xmath4 .
first , we consider the newtonian gravity among three masses denoted as @xmath5 @xmath6 in a circular motion . the location of each mass is written as @xmath7 , where we choose the origin of the coordinates as the common center of mass , so that @xmath8 we start by seeing whether the newtonian equation of motion for each body can be satisfied if the configuration is an equilateral triangle .
let us put @xmath9 , where we define the relative position between masses as @xmath10 and @xmath11 for @xmath12 .
then , the equation of motion for each mass becomes @xmath13 where @xmath14 denotes the total mass @xmath15 .
therefore , it is possible that each body moves around the common center of mass with the same orbital period .
figure [ fig - tri - all ] shows a triangular configuration for general masses .
( [ n - eom ] ) gives @xmath16 where @xmath17 denotes the newtonian angular velocity .
the orbital radius @xmath18 of each body with respect to the common center of mass is obtained as @xcite @xmath19 where we define the mass ratio as @xmath20 .
for the restricted three - body problem , an inequilateral triangular solution was investigated @xcite .
hence , for three finite masses , we study a pn triangular configuration .
let us denote each side length of a pn triangle as @xmath38 where @xmath39 denotes the non - dimensional correction at the 1pn order ( see fig .
[ fig - tri - all ] ) . here , if all the three corrections are equal ( i.e. @xmath40 ) , a pn configuration is still an equilateral triangle , though each side length is changed by a scale transformation as @xmath41 .
namely , one of the degrees of freedom for ( @xmath42 , @xmath43 , @xmath44 ) corresponds to a scale transformation , and this is unphysical . in other words ,
we take account of only the corrections which keep the size of the system but change its shape . for its simplicity , we adopt the arithmetic mean of three side lengths in order to characterize the size of the system as @xmath45 , \label{add}\end{aligned}\ ] ] ( see appendix [ length ] for possible choices of fixing the unphysical degree of freedom ) .
this arithmetic mean of the pn triangle is chosen to be the same as a side length of the newtonian equilateral triangle as @xmath46 = a , \end{aligned}\ ] ] so that the degree of freedom for a scale transformation can be fixed .
otherwise , a degree of freedom for a scale transformation would remain so that an ambiguity due to the similarity could enter our results .
thus , we obtain a constraint on ( @xmath42 , @xmath43 , @xmath44 ) as @xmath47 hence , we look for the remaining two conditions for determining ( @xmath42 , @xmath43 , @xmath44 ) in the following .      furthermore , @xmath53 and @xmath54 also have pn corrections . however , these corrections multiplied by @xmath42 ( or @xmath44 ) make 2pn ( or higher order ) contributions in eq .
( [ m1-eom - c ] ) and hence they can be neglected .
also in @xmath29 , 1pn corrections to @xmath53 and @xmath54 lead to 2pn , since they are multiplied by 1pn term as @xmath55 .
we obtain the equation of motion for @xmath35 and @xmath36 by cyclic manipulations as @xmath37 .
thus , we obtain the expressions for ( @xmath42 , @xmath43 , @xmath44 ) as @xmath60 , \label{d12 } \\
\varepsilon_{23 } & = \frac{1}{24 } \frac{m}{a } [ ( \nu_3 - \nu_1)(5 - 3\nu_2 ) - ( \nu_1 - \nu_2)(5 - 3\nu_3 ) ] , \label{d23 } \\
\varepsilon_{31 } & = \frac{1}{24 } \frac{m}{a } [ ( \nu_1 - \nu_2)(5 - 3\nu_3 ) - ( \nu_2 - \nu_3)(5 - 3\nu_1 ) ] , \label{d31}\end{aligned}\ ] ] which recover previous results for the restricted three - body problem @xcite .
substituting eqs .
( [ d12 ] ) and ( [ d31 ] ) into eq .
( [ m1-eom - c ] ) , we obtain the angular velocity as @xmath61 where @xmath62 .
\label{omega - pn}\end{aligned}\ ] ] by using @xmath63 , one can immediately show @xmath64 , so that we find @xmath65 for the same masses and @xmath66 .
in other words , for the same masses and angular velocity , the pn triangular configuration is always smaller than the newtonian one .
table [ table1 ] shows the relativistic corrections of the distance between each body for lagrange point @xmath2 ( @xmath1 ) of solar system .
here we choose @xmath27 and @xmath35 as the sun and each planet , respectively . in the case of the restricted three - body problem ( @xmath67 ) , it is convenient to use eqs .
( [ condition1 ] ) and ( [ condition2 ] ) rather than eqs .
( [ d12 ] ) - ( [ d31 ] ) , because it is natural to change not @xmath68 but location of @xmath36 .
( [ condition2 ] ) implies that the correction of distance between each planet and lagrange point @xmath2 ( @xmath1 ) is approximately @xmath69 of schwarzschild radius of the sun .
hence , we obtain the same values of this correction for earth and jupiter .
the similar corrections are mentioned also in the previous paper @xcite .
the above pn effects , however , are so tiny that they could be neglected in the near - future measurements @xcite .    it is interesting to extend this 1pn work to higher pn orders for the gravitational wave physics ( see @xcite for the equation of motion and compact binaries ) .
we reexamined the post - newtonian effects on lagrange s equilateral triangular solution for three - body problem . for three finite masses
, it was found that a general triangular configuration satisfies the post - newtonian equation of motion in general relativity , if and only if it has the relativistic corrections to each side length .
it was shown also that the post - newtonian triangular configuration is always smaller than the newtonian one for the same masses and angular velocity . studying the correction to stability of this configuration
is left as future work .
this work was supported in part ( k. y. ) by japan society for the promotion of science , grant - in - aid for jsps fellows , no .
in stead of the arithmetic mean of the three side lengths , one might wish to use the geometric mean of them or the triangular area . hence , let us mention briefly these cases .
the geometric mean of three side lengths by eq .
( [ ep ] ) is written up to the 1pn order as @xmath70 + \mathrm{o}(\varepsilon^2 ) , \label{mul}\end{aligned}\ ] ] where @xmath71 denotes the second order of ( @xmath42 , @xmath43 , @xmath44 ) , namely at the 2pn order .
this expression is identical with the arithmetic mean by eq .
( [ add ] ) .
in addition , we can obtain the condition that these means are equal to a side length of the newtonian equilateral triangle as @xmath72    next , we consider a pn triangular area . a triangular area @xmath73 is given by heron s formula as @xmath74 where @xmath75 hence , substitution of eq . ( [ ep ] ) into eq .
( [ area ] ) leads to @xmath76   + \mathrm{o}(\varepsilon^2),\end{aligned}\ ] ] which corresponds to the triangular area of each side length eq .
( [ mul ] ) .
therefore , a pn triangular area is equal to a newtonian equilateral triangular one if and only if @xmath77 this is identical with the condition eq .
( [ epsilon - total ] ) .
99 h. goldstein , _ classical mechanics _ ( addison - wesley , ma , 1980 ) .
j. m. a. danby , _ fundamentals of celestial mechanics _
( william - bell , va , 1988 ) . c. marchal , _ the three - body problem _ ( elsevier , amsterdam , 1990 ) .
m. connors , p. wiegert and c. veillet , nature * 475 * , 481 ( 2011 ) .
e. krefetz , astron .
j * 72 * , 471 ( 1967 ) .
t. i. maindl , _ completing the inventory of the solar system , astronomical society of the pacific conference proceedings _ , edited by t.w .
rettig and j.m .
hahn , * 107 * , 147 ( 1996 ) .
h. asada , phys .
d * 80 * 064021 ( 2009 ) .
n. seto , t. muto , phys .
d * 81 * 103004 ( 2010 ) .
j. d. schnittman , astrophys .
j. * 724 * 39 ( 2010 ) .
y.  torigoe , k.  hattori and h.  asada , phys .
.  lett .   * 102 * , 251101 ( 2009 ) .
k. nordtvedt , phys . rev . * 169 * 1014 ( 1968 ) .
t. ichita , k. yamada , h. asada , phys .
d * 83 * , 084026 ( 2011 ) . c. w. misner , k. s. thorne , j. a. wheeler , _ gravitation _ , ( freeman , new york , 1973 ) .
l. d. landau and e. m. lifshitz , _ the classical theory of fields _
( oxford , pergamon 1962 ) .
h. asada , t. futamase , p. hogan , _ equations of motion in general relativity _
( oxford university , new york , 2011 ) . ref .
@xcite suggests that if two of the masses are finite and equal , and the third one is zero , an equilateral configuration satisfies the eih equation of motion .
however , the angular velocity of each mass is not the same in this case .
note that this case is not an equilibrium configuration .
k. yamada , h. asada , mon . not .
. soc . * 83 * , 084026 ( 2011 ) .
t. damour , p. jaranowski , g. schfer , phys .
d * 63 * , 044021 ( 2001 ) .
y. itoh , t. futamase , h. asada , phys .
d * 63 * , 064038 ( 2001 ) .
k. g. arun , l. blanchet , b. r. iyer , m. s. s. qusailah , phys .
d * 77 * , 064035 ( 2008 ) . y. itoh , phys .
d * 80 * , 124003 ( 2009 ) .
.the corrections for lagrange point @xmath2 ( @xmath1 ) of solar system .
( [ condition1 ] ) and ( [ condition2 ] ) are used for the evaluation . here , we choose @xmath27 and @xmath35 as the sun and each planet , respectively .
thus , @xmath80 is the distance between the sun and each planet . [ cols="<,<,<,<",options="header " , ]
continuing work initiated in an earlier publication [ ichita , yamada and asada , phys . 
 rev . 
 d * 83 * , 084026 ( 2011 ) ] , we reexamine the post - newtonian effects on lagrange s equilateral triangular solution for the three - body problem . for three finite masses , it is found that a triangular configuration satisfies the post - newtonian equation of motion in general relativity , if and only if it has the relativistic corrections to each side length . 
 this post - newtonian configuration for three finite masses is not always equilateral and it recovers previous results for the restricted three - body problem when one mass goes to zero . 
 for the same masses and angular velocity , the post - newtonian triangular configuration is always smaller than the newtonian one .