N Body problem with equal masses

                                                     centralpos [1]
I. Kotsireas, Ilias.Kotsireas@lip6.fr

Polynomial systems arising in the study of central configurations in the Nbody problem of Celestial Mechanics. See \underline{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 \underline{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 \underline{B. Mourrain}

Using the maple package \underline{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 s^{15} (54 s^{9} + 243 s^{8} − 270 s^{7} − 1350 s^{6} + 450 s^{5} + 1080 s^{4} − 250 s^{3} − 450 s^{2} + 125)

(26244 s^{43} + 288684 s^{42} + 575181 s^{41} − 7160238 s^{40} − 50847021 s^{39} − 60982308 s^{38}

+660835827 s^{37} + 3076587522 s^{36} + 1456109865 s^{35} − 26462045196 s^{34} − 72545242572 s^{33}

+36782403912 s^{32} + 471517095996 s^{31} + 472035267792 s^{30} − 1454054140404 s^{29}

−2797345375128 s^{28} + 2657399034204 s^{27} + 7314994875672 s^{26} − 3287306912490 s^{25} −10745467046244 s^{24} + 3006653611050 s^{23} + 8621899509096 s^{22} − 1812423397446 s^{21}

−2102960995236 s^{20} − 86038892226 s^{19} − 2565638766072 s^{18} + 1704248798580 s^{17}
+2418961945320 s^{16} − 2068526855700 s^{15} − 250219185840 s^{14} + 1410376311900 s^{13}
−839763655800 s^{12} − 620243819400 s^{11} + 654068946300 s^{10} + 173872733125 s^{9}

−222639990750 s^{8} − 25857433125 s^{7} + 25224697500 s^{6} − 321103125 s^{5} + 7461731250 s^{4}

+1557890625 s^{3} − 3005437500 s^{2} − 911250000 s + 675000000)
(3 s^{2} − 6 s − 5) (s − 1)^{24} (s + 1)^{24}

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] D. Bini and B. Mourrain. Polynomial test suite. 1996.