The existence of global solution of the n-body problem

Chin.Astron.Astrophys.10 (1986) 1x5-142 Act.Astron.Sin.26 (1985) 333- 342

THE EXISTENCE

WANG

Qiu-dong

OF GLOBAL

SOLUTION

Pergamon Journals. Printed in Great Britain 0275-1062/86$10.00+.00

OF THE N-BODY

PPQBLEM

Department of Astronomy, Nanjing University

Received 1984 May 11

ABSTRACT BasedonMcGehee's transformationf=r-lq; g=r1/2p; dt/dt' =r312 , I introduce the transformation x= (g%-lg) 112, h,O X= (g%+g- 21h)l", h.50, I prove that these variables may be continued to every point of the new time axis r for any initial value, and the whole axis corresponds to the "time interval of existence of the global solution". Also, F, G, u are o(eBT). I then obtain a region H on the complex plane r, IIm(T)I
1.

INTRODUCTION

The question of global solution is a basic one in the time-honoured topic of the motion of n bodies. For the J-body problem, by means of the "regularization"technique and a series of detailed estimates, Sundman [l] obtained the method that gives global analytic solutions of the coordinates in the case of C#O, thereby affirming the global integrabilityof this problem. However, his series solution has no practical value because of their convergence being very slow. Because of its inherent limitations, this method cannot be generalized to the general, n-body problem, and is not applicable to the case C= 0 of the 3-body problem. Up to now, no real progress has been made in these directions. Over the last twenty years, Pollard [2], Saari [3], and Marchel and Saari [4] have more or less completed the work of estimating the Sundman type parameters in the n-body problem. These authors have generalized almost all the qualitative conclusionsobtained in the 3-body case to the n-body case, thus providing the pre-conditions for a consideration of the global solution of the n-body problem. Thanks to a large body of theoretical and numerical research [S, 61, we now have a basic understanding of the various types of

singularities in the motion of n bodies, and the kinematic behaviour in the approaches to these singularities. The result of the discussion has almost completely negated the possibility of "regularizing" triplecollision singularities [7]. A recent study [8] has further proved the existence, under the premise that allows for regularization of 2-body collisions, of non-collisional singularities for which we are totally unable to apply the concept of "regularization". Meanwhile, efforts at improving the convergence of the Sundman theory have not made any progress. These results force us to consider the implications of the Sundman method on the n-body problem from a negative point of view, and suggest that we should tackle the problem with the help of the recent results in the studies of kinematic morphology. In this paper, by improving the McGehee transformation,a new set of variables is introduced to overcome the difficulties caused by the singularities and a specific method and procedure is given for finding a global solution of the n-body problem.

136

WANG

2.

NOTATION.

1.

Definitions

CONVENTIONS. PRELIMINARIES, and Conventions

Starting from any given initial Definition. point, the largest analytically continued solution of the dynamical system is called a “global solution” of the system. It is also called a “motion” of the system. The largest continuable region of the independent variable is called the “time interval of existence of the motion”, and the motion is said to “terminate” at the end of the time interval of existence. Because of the reversibility of the n-body problem, we shall discuss only the positive time direction, and merely state the result for the negative time direction. Let

where VU(a) is the with respect to q. following notations:

R = l,l,“;{r;,),

2.

min{r,,}, l-e/

r -

Properties

gradient We also

of

l’-

UGA(tlor

as t+tl,

t)-2/3,

2) non-collision

singularity,

position and let

and momentum vectors

I r, - r,l = i(x, - x,)’ +

r,j -

(y,

-

y,y

+

of

the

U>Al(tl

- t)-2/3.

(ii) infinity,

If

(2‘ - -,>‘Y’”

be the distance between mass-points i and j. The potential function of the n-body system is

and its

kinetic

The energy

energy

integral

the motion then,

the

terminates

system

,W -

the

energy

constant.

diag(m,, nt,, m,; %¶ m,, m,;.-*;

We then

at

sufficiently

I-$,

[9]

g = I”‘&

introduced

there

time

large

t,

the

following

dt/dr’ = I”‘,

If q(t), p(t) are a solution (l), then the corresponding solution of the equations.

is

(2) of

f,

equations 4, I are a

(f, g)l,

(3) (4)

(f, gX -+- M-‘g,

The prime in (3) - (5) denotes d/dt', At the same time, the following equations are automatically satisfied

Let m, ,m*, pts.>

f=~f -

1;

T(~) -

U(l) -

16.

(7)

have 3.

The equations

ej- M-‘9,

tl,

8’ = $ (f, gig-I- VU(f), di/dr' - I”‘,

T-U-h where h is

f -

near

The McGehee Transformation

In 1974, McGehee transformation:

1’ f’--

is

for

3.

where,

- Q213),

(h > 0) , or :ziie2/3) A ‘Ef/3for (h > 0). be the system

qTMq,

Evolution

i.e. when t is sufficiently exists A1 >O such that and

U(q)

Pollard, Saari, and others have discussed in depth the evolutionary properties of the n-body motion. Here we list those results that are relevent to the paper. (i) If the motion terminates at a finite time tl, then either of the following cases must obtain: 1) collision singularity, where

R-tco, r=O((tl

be the position vector, momentum vector Let mass of the i-th mass-point.

vector of have the

+-

of

VV(S).

the

n-body

motion

are

A NEW TRANSFO~ATION AND ITS PROPERTIES

I transform the again and put

McGehee transformation

once

Global

x - (gTM_‘g)“‘, X = (gTM-‘g The relations

It

is

between

u-’ -

2(U(q)

u-’ -

2U(q)

easily

the

that

= M-‘G

dG/dr

= VU(F)

drldi

= P.

The following

,

equations G’M-‘G

I GTM-‘G

F -

G -

x-lg,

at’jdr

G -

-

x-3.

dr/dr

set

the

(811

h GO.

r and the

G,

u”‘p,

satisfy

t

h>O,

X’j,

u, F,

i(_‘q,

G,

-

P.

of

(8)2

original h 2

variables

4,

are

(9) 1 (912

equations, (10) (11) (12) (13)

VU(F))G,

automatically -

1

2

f

1 + Zuh,

-

U(F)

satisfied: -

h > 0,

uh

FrM F -

U(F)

t are

0,

VU(F))F,

(M-‘G,

p,

h < 0.

VU(F))u, + 2(&f-%,

= 1 -

1x-1,

F=

u, F,

= -2(M-‘G,

dF/ds

u -

new variables

+ h)

derived du/di

2Ih)“‘,

137

Solution

-

(14)l

I’u-‘, h < 0.

0,

(14)2

A solution of the new set of equations, with as initial values the ug, Fg, Go, TO derived from some pa, 40, to according to the transformation, constitutes a solution of the n-body motion. Since the two equations (11) and (12) are separate from the two equations (10) and There is no essential difficulty to we shall, for simplicity, discuss only F and G. (13)) apply the following argument to a combined discussion of all four variables F, G, u and t. Lemma 1 (i) If the motion of the original equations can be continued to t=m, then t+-COmust correspond to T+W. T can be continued to infinity, then r-+m (ii) If for a solution of the new equations, corresponds to the corresponding solution of the original equations with t tending either to In the latter case, the original motion terminates at tl. infinity or to a finite value tl. (iii) If the motion of the original equations terminates as t+tl, then we must have lim r ==.

t-e1 (iv) analytic Proof in the

The value is finite.

of

(i) If in the case of h&O,

r corresponding

to

solution of the for sufficiently

original large

k j’ (u + h)“‘du ’ =

1

a point

4 j‘ h”‘d,r

I( j’ ““‘du

r= can

only

Lemma 2

be

diverges, solution

t-axis

at which

the

solution

is

equations, t can be continued to infinity, exists B such that r5B td3. Hence

then,

t, there h>U

z I i&f cr-‘drr me O3 h < 0

(ii) If -r+m, then either ~+m (from the then either t+m or t+tl is a singularity. (iii) If the motion terminates as t-t tl, collision singularity occurs at tl. When t and -1, U “‘du 2 ,I , 5” [ (,, - rc)--“‘]“‘,,jrr = r1 The right side (iv) If the hence

on the

hence T+-. is analytic

at

time transformation relation) or t+m. QED. then either a collision singularity or is sufficiently near tl, we have U>A(tl .I, i” (I,

tl,

then

value

of

If

u+m,

a non- t)2/3,

rr)-‘du.

--.

u is

finite

as

t-+tl.

Also,

tl

is

finite,

solution

r of

” U”‘drc 5 finite

Corresponding

to

the

initial

the

original

equations,

the

the

138

WANG

new equations must be continuable to infinity. Proof

From (14),

!--

(15)

-2 1 1 - U(F) U(F)-= 0, 2uh,

h h <2 IV, II.

(16) (17)l (18) Also, GTM--'G= 1,

h 2 U:

G+M-‘G=i+Zuh,

AiV.

Hence !Cils&(m),

.4,(m) - (iii)".

(17)z

Suppose the motion cannot be continued to r+. following cases must appear: i)

I:

*CU;

Ii)

iii) r(F)

6*co:

-

l;$j(F,j(l:l\

Then as T-+TO (finite), one of the -*@*

Obviously, only i) can happen, i.e., F+m. In this case, either I+- or U(q)+m will correspond to the time of termination. From Lemma 1, TO cannot be finite, and we have a contradiction. Summarizing the above, we have Theorem 1 Every global solution of the original equations corresponds to a global solution of the new equations with time interval of existence +:(-m, +m). Analytic time instants t of the original solution correspond to finite time instants of t.

4.

ANALYTIC REGIONS OF F AND G

I discuss further F and G. NOW, T will be regarded as complex, so all the continuations in the previous section refer to the real axis of the complex plane. It was proved that the solution can be continued to infinity along the real axis of r. Therefore, for each point '0 of the real axis, a time neighbourhood exists, within which the motion is analytic. It then follows that there existsa region containing the whole real axis, over which the solution is analytic. To find a global solution, we must identify this analytic region. Theorem 2

For any real T and any F'i(r),we have

j F,(f)j < (I/&)(/1,

+ A,/ F,(O)l)e”*” - &A.

where _I,= 3nM-"'/4m', Al= y/&z.

(19)

Theorem 3 Regarding r to be complex, a solution starting from an initial value can be continued to any point 70 on the real axis. Moreover, it is analytic in a region /z - T"j< C/(,4+ B,r""*') where A, B, B1, c are constants depending on the masses and the initial value. Proof That the solution can be continued to any point ro on the real axis was proved in the last section. For to, according to (14), we have r,,(F(r,)) Z iJ,(nl);

1G,(r,)

1 -G /J,(m).

Take the following neighbourhoods of ~(?a) and G('r0): P:

We have

lc, - Gt(dJ;

IF, - F,(r#)l < min(.<,, zi,)/+dy- ifa.

120)

Global

= irdfo) -

I(r, - r,)(F)1

ri(fo)

[r, -

+

139

Solution

r,(rdl

+ [r,(d

- rill

Z ri,(d - [ lr, - r,(6) I + I r, - r,(s) 1I.

while /r, -- r,(r,)l(I:)

= I(/;,,

.-’ 1:,,(f,,))‘+

(F,,--

I;,,(ro))’ + (F,; -

F;;(fo))‘]“‘

< (3.1:)‘” -

J 3 ,l‘.

Similarly

1r, -

r,I (F)

2 A, -

2J3Ab

2 A, -

2J3

. A,/4&

-

$ A,.

Also

For the

right

side

of

the

equations

for

M -

A7 + A,1 r,,(s,)

A, -

6n(A, + A&) * 4ti=/A:m,

where

A7 _ 4$/A:

F and G, we have

the

upper

bounds

I , I I:,“(G) I - llIPX{I Fi(d I j.

+ (A, + A,)/m

+ 3n(A, + AG)’ * @‘/Aim

(21) (22)

-f AdA..

while Ik’,(r,))

<(A,

+ A,IFi(O)l)

. A;‘c”a”J - A,/4

< (l/A,)(A,

+ A,(F,(O)))&‘.‘.

where F,.(U) - maxi I F,(O) 11. According

to 2:

over

which

Collecting

the

5.

above,

there

exists

a neighbourhood

B = A,,

where B, -

(30 + l)(A,

+ A,F.(O)

* AI/A,),

C-

axis r over specifically

which

Ar.

we have

exists initial

Il,,,(f)l

Theorem,

+ B,cW’).

analytic,

3(n + 1)A,,

There given H:

in which

Existence

15 - foI < C/(A

F and G are A -

Theorem 4 satisfying

Cauchy’s

a neighbourhood H of the real values exist and are analytic;

< C/(A

A, B, B1, c are

solutions

F and G

(23)

+ BIeB’““).

as given

for

above.

ANALYTIC MAPPING

If we can find an analytic mapping, which maps region H of the then of H containing the whole real axis onto the unit circle, found by expanding along the new complex variable. We shall discuss only analytic mapping and shall freely use that have been used before as long as no confusion arises. Lemma

There

Proof

We have

exists

H,:

II,,,(r)I
H,cH

where Aq is

complex t-plane or the global solution some of

the

algebraic

a constant.

C/(A

+ B,,a’wcr’) > C/(,4 + B,)cW'LI'l _ [C/(A + ~,)]e-~IWP . eBW)'--Illl.'l

A, -

[C/(A

> [C/(A + B,)]

Obviously, H,CH. Introduce conformal

+ B,)]

. e-b

a subregion can be

. c-B(=‘)’ _ A,r-“Rt=“‘,

(24)

. e-is*

mapping

f: t-X+iY--+z==u+j~ 5: z=fe 6'.

(25)

symbols

WIG

140

Y) -iv(X, Y)i,

z -

u(X,

I4-

P’~‘-‘“(Xcos(2XY)

-

“-r”‘X’~~“(Ycos(ZXY) Let

&af be

Y = _+a eeBx2,

v(l.f) Take a to

the

always

0.

Their

images

by

6z* emax * cBx’-“cxP’-zsx*’ - cos(2aX

in

f

z are

* e-sx’)

+ X sin (20X

. ,-.X1) _ (I) + ([[),

be

(I -

since

a >

Ysin(ZXY)),

+xsin(2xY)),

min

{J--ET% -

* 7,

2

B, l/[ZS

+ (B + l)‘fZ]“,

maximum value of x* e-BX’2 is e -aQrpl-anx’),

c ..I’ ;

l/Zd(Be),

‘4,

I

we have

Za.X.e-BX2
Also, we have

X . sin(2uX . e-sx’) > 0

Hence, U(lZ) -

(I) + ([I) > (t) > (I * e--+COS

(26) Similarly, Denote

we have vi!& < -Ali). the neighbourhood of the real

axis

bounded

by ~2 by Xp.

Lemma 3 Define in the z-plane, region 6: /IBIZ/
that, for x fixed, and (a as shown above).

e“‘X’-r’[-ZBY’cor(2XY)

-I- cos(2XY)

_ eB(x’-Y’)[ (1 _ 2BY5lcos(2XY) -

-

increasing

-I- 2X’cos(2XY)

ZBYXsin(2XY)

H&H,.

-

from

there a I-

0 to a exp -BX’,

-i- 2x1 cos(2XY)

-

exists 1

v@, Y)

?(B + l)XYsin(2XY)J

2XYsin(2XY)]

,*tx’-Y’J[(I) + ((1) I- (U[)l.

Since, with the above we have 1 - 2uya 7 I -

2tlc-‘~xtlra, and since

(0

2&U -

+ ([I) + ([If) > [I -

p < flI2d:

Y

Obviously,

We havej(Hj)3g. Further, that f(H3) =H expresses

choice

-I- (B + i)‘/2]“‘<

of

a, j2XYj

< /2X.

o~-BX’/c rr/4 , we have (11) > 0; (W) < 0. As for

B . 2(B + I)/Xj + 2x’]y/?/2.

-?..-.,

we have (I) 7 0.

!a?e have

as Y varies from 0 to negative values, Similarly, for a fixed Given any (x,Y) 6 I$, since v(&;) > Alo, (see Figs. 1 and 2). a$, such that v(X, Y(X)) = y.

d&X,

P < J/i28

+ (R + l)‘/2Pt,

(1) -+ (If) + (ILL) 7 0 always,

we have hence

J2B

av/au> 0 always.

Step 2 To prove U(X0, Y(X0)) = x.

Since

that, Y(X))/dX

for

such

-&f&X

a Y(X),

there

exists

X,

we also there a unique

have av/aY> 0. must be a Y(X), point

~0

such

uniaue

within

that

+ (au/au). (dYfdX)

= aMfax f (a,faYji(- adaxh4Wa~~l

= au/ax + (aufaY)~(a,iaY)f(a~fax)l w [(allfax)'+(~~faY)zif(a~f~y) =-u The second line is because V(X, conditions of analytic functions. completes the proof of Step 2.

Y(X))

= y, the third line follows from the Moreover, ~(0, Y(O)) = 0, and u(X, Y(X))

Cauchy-Reimann +m as x-f-.

This

C,

Global

Thus, any point (x, y) in E corresponds This to one and only one point (X, Y) in Hz. mapping at the same time maps the real axis The original image of ii onto the real axis. in Hi must be a neighbourhood of the entire real axis.

Fig.

is a single-valued, analytic branch of which can be found through the inverse f-l> function of f (all the series coefficients to be real). Now apply the Poincarc! mapping, y:z+S,

‘,’ = g’,j.

II, -+ 8,

,,,-’ = T-‘“k-‘,

R of the mapping, Q-

]I,,

Under and this is a 1 - 1 analytic mapping. 4, the entire real axis of r is mapped onto the interval (-1, 1) on Q, and the search for the mapping is completed. From the foregoing discussion we have the following theorem on global solutions: Theorem 5 F(T) and G(r) can be expanded as power series in the new independent variable The series converge on the unit circle S. of the s-plane and the real axis interval (-1, 1) of s corresponds to the time interval of existence of global solution in the phase space of the n-body motion. We have now given a method of global integration of the n-body motion, thereby affirming the integrability of this problem in the sense of analysis.

6.

Write

141

this

mapping

(1, 7-1,: fl,+-+H,

DISCUSSION ON METHOD

Global solution of dynamical system should be understood as the largest continued Singularities are solution of the system. the key difficulties in the finding of The existence of globular solutions.

and its

inverse

J(If,) - R;

f_‘(R)

I-

K?W))

Fig.

I

fl

y maps ii onto the unit circle Take the composite S-plane.

Solution

I;

as, - II,;

2

singularity deprives us from knowing just how far forward (or backward) we can continue. The whole process of finding global solution is essentially a process of overcoming the difficulty of singularities. Sundman first used the sufficient condition for total collision to exclude the possibility of total collision of three In this way he circumvented the bodies. difficulty of triple collision, though at the expense of narrowing down somewhat the range of applicability of discussion. Next, and of key importance, he altered the concept of global solution of a dynamical system. By means of the “regularization” technique, global solution of motion was understood no longer as the largest continued solution of the “phase but as the largest continued space”, solution of the “position coordinates” with He replaced respect to a new time variable. the classical continuation of “phase space coordinates relative to the independent variable by the analytic continuation of “position coordinates” relative to a new At the appearance of independent variable. a two-body collision singularity, because of its “removeability”, the “position coordinates” can be continued indefinitely with respect to a “regularized” time variable, right to infinity in the original time, thus overcomine the difficulty of the sineularitv. In the-past it was thought that global . solutions are solutions valid for the entire time interval (-m, +m). This is an illusion. Global solution, in the Sundman sense, is the largest continued solution of the “position coordinates”, and it is only in the case of binary collision that such a solution can exist over time (-m, +m). If the

WANG

142

collision is of a more general type, then just how long such a global solution can Of course, exist will be difficult to say. the hope has been that it does exist, thereby solving the difficulty of singularity Researches over the in the n-body problem. last hundred years have essentially given a It has been found that the negative answer. singularity of multiple collision is “inherent” and not removeable, which continuation in the Sundman sense cannot pass

In other words, there is no way of through. overcoming the singularity difficulty. The transformation given in this paper<' has the following property: it makes every largest continued solution's time interval of existence to correspond to the interval. f,m -t*)in the variable T. Global sol&ions faund by the method here revert

to their original "%xitural" meaning. The key idea in this paper is to unify the sise of the time interveil of existence before Fwthermore, continuing the saluliun. because of the properties of the equations (10) - (1X), the process of estimating the analytic region is very simple. The series solution given here likewise To solve this suffer from slow convergence. of the problem, an in-depth discussion properties of complex singularities is This is a difficult and indispensable. interesting problem.

~CK~~WL~~G~~~E~T The author thanks Professor VI Zhao-hua, De~ar~meni of astronomy, Nyjing University, for help and guidance in the present

work

t

REFERENCES Sicgcl C. L; M~brr J. K: Lzcturron celestial mechmics, 1971 Springe. PollardH: Gravitational system, J. Molt. and Mech, 17( 1967), 601-612. Saari D. G: Expanding &wiratinnal system, Trans. Amer. M&h, SW, 156( 1971) 219-240. Marchel C; Saari D. G: On the final evolution of the n-bcdy problem, j. I.Xfi.I?+, 20( 1976). 15ft186.