This page is served as a supplement of my joint work with Ke-Ming Chang [1,2] on finiteness of central configurations for the planar six-body problem by symbolic computations.
In [1,2] we develop symbolic computation algorithms to investigate finiteness of central configurations for the planar n-body problem. Our approach is based on Albouy-Kaloshin's work on finiteness of central configurations for the 5-body problems (Ann. Math. 2012). In their paper, bicolored graphs called zw-diagrams were introduced for possible scenarios when the finiteness conjecture fails, and proving finiteness amounts to exclusions of central configurations associated to these diagrams. Following their method, the amount of computations becomes enormous when there are more than five bodies. In our work we introduce matrix algebra for determination of possible diagrams and asymptotic orders, devise several criteria to reduce computational complexity, determine possible zw-diagrams by automated deductions, and generate mass relations associated to many diagrams through symbolic computations. For the planar six-body problem, we show that there are at most 85 zw-diagrams, 61 of them are impossible except for masses in a co-dimension 2 variety in the mass space.
A preliminary version has been posted on arXiv. It is quite long, so we split that into two papers [1,2] to make it more accessible and readable. Here we post a separate and updated Mathematica worksheet for readers. There are three algorithms, the first one is to find zw-matrices, the second one determine orders of variables, and the last one generate mass relations. Our first algorithm narrows down the case of 6 bodies to 117 zw-diagrams, the second algorithm eliminates 31 of them, the third algorithm eliminates one more, leaving 85 possible diagrams, and eliminate 61 diagrams among them by finding mass relations. This leaves 24 unsolved diagrams. We have combined all of them into one worksheet:
The file contains instructions for the usage, side notes showing correspondence between subroutines and criteria been applied, as well as several examples discussed in [1,2]. A technical merit of our work is providing a convenient platform for this research topic via symbolic computations -- by exploring more and more criteria or rules for central configurations, one can simply transform them to new subroutines and add them to our algorithms. It is likely that finer or more advanced computational tools may generated mass relations for some of our unsolved cases.
References
| Last modified: July 31, 2023 |