Numerical Solutions for N-body problems


The isolated real solutions of some Albouy-Chenciner equations are available for download.  Unlike random search by Newton’s method, HOM4PS-2.0 computes all isolated solutions by the polyhedral homotopy method.   In addition, all solutions are verified by using the interval Krawczyk method (with radius 10^-8) to guarantee each exact solution lies in the interval of the numerical solution with radius 10^-8.  Hence, they yield at least 7 digits of accuracy.  We use the interval arithmetic in INTLAB (a Matlab toolbox with interval arithmetic) to implement the verification. The verification results are also available for download.



3-body problem


(m1,m2,m3) = (1,1,1)       7 real solutions    4 physical solutions    verification



(m1,m2,m3) = (1,2,3)       7 real solutions    4 physical solutions    verification



4-body problem             HOM4body   (download)


(m1,m2,m3,m4) = (1,1,1,1)      135 real solutions      32 physical solutions    verification



(m1,m2,m3,m4) = (1,1,2,2)      139 real solutions      24 physical solutions    verification



5-body problem


(m1,m2,m3,m4,m5) = (1,1,1,1,1)     8775 real solutions    258 physical solutions    verification



(m1,m2,m3,m4,m5) = (1,1,1,1,2)



6-body problem


(m1,m2,m3,m4,m5,m6) = (1,1,1,1,1,1)




1.      Physical solutions are real solutions with positive mutual distances.

2.      For getting more accurate digits, Newton’s method with multi-precision can be applied.






[1] T. L. Lee and M. Santoprete (2009), “Central configurations of the five-body problem with equal masses”, Celest. Mech. Dyn. Astr., 104, pp369-381. (On-line)






Contact information:

Tsung-Lin Lee (leetsung@math.nsysu.edu.tw)

        Manuele Santoprete (msantopr@wlu.ca)