HOM4PS-2.0

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)

Note:

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.

Reference:

[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)