N Body problem with equal masses

centralpos [pol96]

I. Kotsireas, Ilias.Kotsireas@lip6.fr

Polynomial systems arising in the study of central configurations in the N-body problem of Celestial Mechanics. See here for more information.

Characteristics:

Dimension 0.

Example 1:

# 4 bodies with equal masses, before symmetrisation.
# We are looking for the real positive solutions
vars:=[B,D,F,b,d,f]

[(b-d)*(B-D)-2*F+2,
(b-d)*(B+D-2*F)+2*(B-D),
(b-d)^2-2*(b+d)+f+1,
B^2*b^3-1,D^2*d^3-1,F^2*f^3-1];

Example 2:

# 4 bodies with equal masses, after symmetrisation
# We are looking for the real positive solutions

vars:=[s,p,S,P,TH,F,f];

[2*Theta-S*s-2*F+2,
8*F*p-4*p*S-2*F*s^2+S*s^2+4*Theta-2*S*s, -2*s-4*p+s^2+f+1,
s*Theta^2-p*s*P-p*S*Theta-2, p^3*P^2-1, F^2*f^3-1];

Example 3:

# S4: 4 bodies with equal masses, after fundamental theorem 
# of symmetric functions.
vars:=[s,p,phi];

[-2*p**3+2*p**3*phi**3-4*phi**3*s*p**2+5*phi**3*s**3*p-phi**3*s**5,
 -2*s*p**3-2*phi**3*s**2+phi**3*s**4-3*phi**3*s**2*p+2*phi**3*p,
 -2*s**2+s**4-4*s**2*p+phi**2+1+4*p];

Example 4:

# 5 bodies with equal masses, before symmetrisation.
# We want the positive real solutions. 

vars:=[B,D,F,b,d,f];

[3*(b-d)*(B-D)+(B+D)-6*F+4,
3*(b-d)*(B+D-2*F)+5*(B-D), 
3*(b-d)^2-6*(b+d)+4*f+3,
B^2*b^3-1,D^2*d^3-1,F^2*f^3-1];

Example 5:

# S5: 5 bodies with equal masses, after fundamental theorem 
# of symmetric functions.
vars:=[s,p,phi];

[-6*p**3+4*p**3*phi**3+15*phi**3*s**3*p-3*phi**3*s**5-12*phi**3*s*p**2
-3*phi**3*s*p+phi**3*s**3,
-9*phi**3*s**2*p-5*phi**3*s**2-6*s*p**3+3*phi**3*s**4+5*phi**3*p, 
-12*s**2*p-6*s**2+3*s**4+4*phi**2+3+12*p];

Example 6:

# N bodies with equal masses
vars:=[k,B,D,F,b,d,f];

# k is the dimension of the euclidean space. 
param:=[k];

[k*(b-d)*(B-D)+(k-2)*(B+D)-2*k*F+4,
 k*(b-d)*(B+D-2*F)+(k+2)*(B-D),
 k*(b-d)^2-2*k*(b+d)+2*(k-1)*f+k,
 B^2*b^3-1,D^2*d^3-1,F^2*f^3-1];

Example 7:

# 3+1 bodies, before symmetrisation
# m is the value of the mass of the first body 
# the other values are normalized to 1 

vars:=[m,B,D,F,b,d,f];

[(B+D-2)*m^3-((b-d)*(B-D)-2*F+2)*m^2+(2*F-(B+D)-(b-d)*(B-D))*m,
(B-D)*m^2+((B+D-2*F)*(b-d)+(B-D))*m,
(b-d)^2-2*(b+d)+f+1,B^2*b^3-1,D^2*d^3-1,F^2*f^3-1];

Example 8:

# 3+1 bodies, before symmetrisation
# m is the value of the mass of the first body 
# the other values are normalized to 1 

vars:=[m,B,D,F,b,d,f];

[(B+D-2)*m^3-((b-d)*(B-D)-2*F+2)*m^2+(2*F-(B+D)-(b-d)*(B-D))*m,
(B-D)*m^2+((B+D-2*F)*(b-d)+(B-D))*m,
(b-d)^2-2*(b+d)+f+1,B^2*b^3-1,D^2*d^3-1,F^2*f^3-1];

Example 9:

# 3+1 bodies, after symmetrisation
# m is the value of the mass of the first body 
# the other values are normalized to 1.
vars:=[m,s,p,phi];

[-s^3*m*phi^3+s^3*phi^3-phi^3*s^5+3*p*s*m*phi^3-3*p*s*phi^3-4*phi^3*p^2*s
+5*phi^3*p*s^3+2*p^3*m*phi^3-2*p^3,
-phi^3*s^4-p*phi^3-p*m*phi^3+2*p^3*s+3*phi^3*p*s^2+s^2*phi^3+s^2*m*phi^3,
s^4+4*p-4*p*s^2+1+phi^2-2*s^2];

Problem: Compute the positive real solutions.

· Solution by J.C. Faugère

Modular computation of a lexicographic gröbner basis by FGB on PC Pentium Pro 200Mhz, with 128M Ram:

4 Bodies	5 Bodies 1	5 Bodies 2	5 Bodies 3	5 Bodies 4
3.8 s	4.2 s	38 s	18.05s	15.9s

· Solution by B. Mourrain

Using the maple package multires, for the system S5 (5)

S := [-6*p**3+4*p**3*phi**3+15*phi**3*s**3*p-3*phi**3*s**5
-12*phi**3*s*p**2-3*phi**3*s*p+phi**3*s**3,
-9*phi**3*s**2*p-5*phi**3*s**2-6*s*p**3+3*phi**3*s**4+5*phi**3*p, 
-12*s**2*p-6*s**2+3*s**4+4*phi**2+3+12*p];

read multires;
B := mbezout(S,[p,phi]):
T := ffgausselim(B):
factor(lasts(T));

we obtain the equation in s,

1242138237552981939585024 s¹⁵

(

54 s⁹+243 s⁸-270 s⁷-1350 s⁶+450 s⁵+1080 s⁴-250 s³-450 s²+125

)

(	26244 s⁴³+288684 s⁴²+575181 s⁴¹ -7160238 s⁴⁰-50847021 s³⁹-60982308 s³⁸

+660835827 s³⁷ +3076587522 s³⁶+1456109865 s³⁵-26462045196 s³⁴ -72545242572 s³³

+36782403912 s³²+471517095996 s³¹ +472035267792 s³⁰-1454054140404 s²⁹

-2797345375128 s²⁸+2657399034204 s²⁷+7314994875672 s²⁶-3287306912490 s²⁵

-10745467046244 s²⁴+3006653611050 s²³+8621899509096 s²² -1812423397446 s²¹

-2102960995236 s²⁰-86038892226 s¹⁹ -2565638766072 s¹⁸+1704248798580 s¹⁷

+2418961945320 s¹⁶-2068526855700 s¹⁵-250219185840 s¹⁴+1410376311900 s¹³

-839763655800 s¹²-620243819400 s¹¹+654068946300 s¹⁰ +173872733125 s⁹

-222639990750 s⁸-25857433125 s⁷ +25224697500 s⁶-321103125 s⁵+7461731250 s⁴

+1557890625 s³-3005437500 s²-911250000 s +675000000

)

(

3 s²-6 s-5

)

(

s-1

)

(

s+1

)

Among all these factors, we are interested in the factor of degree 43. The function fsolve of maple gives the following real roots:

- 4.312711122, - 4.146301330, - 2.972967371, - 1.953169828, - 1.159781906,

1.647140654, 3.007928062.

Two are positive. Once we have these values, we substitute in B, compute its kernel, and deduce the other coordinates of the roots.

B1 := subs(s=1.647140654, evalm(B)):
evalf(Svd(B1,U));
v1 := evalm(col(U,18)/U[1,18]);
p= v1[2], phi= v1[11];

p= .5816954601, phi = -.8879732985.

One of the coordinate is negative.

B2 := subs(s=3.007928062, evalm(B)):
evalf(Svd(B2,U));
v2 := evalm(col(U,18)/U[1,18]):
p= v2[2], phi= v2[11];

p = 2.126221965, phi = 1.661288273

This is the positive real solution (with 5 digits of accuracy).

References

[1]: R.B. Kearfott. Some tests of generalized bisection. ACM Transactions on Mathematical Software, 13(3):197--220, 1987.

This document was translated from L^AT_EX by H^EV^EA.