Central configurations in the spatial n-body problem for $$n=5,6$$ with equal masses

Abstract

We present a computer assisted proof of the full listing of central configurations for spatial n-body problem for $$n=5$$ n = 5 and 6, with equal masses. For each central configuration, we give a full list of its Euclidean symmetries. For all masses sufficiently close to the equal masses case, we give an exact count of configurations in the planar case for $$n=4,5,6,7$$ n = 4 , 5 , 6 , 7 and in the spatial case for $$n=4,5,6$$ n = 4 , 5 , 6 .

Toward finiteness of central configurations for the planar six-body problem by symbolic computations

“…Do [Do[kk3 = kk3 * Length[zomatrix3[[i3,j3]]] * Length[womatrix3[[i3,j3]]], {i3, j3, n}], {j3, 1, n}]; Return ... ,Bold,16]]; zo1 = Table [{1, 2, 3, 4, 5}, {i1, 1, n}, {j1, 1, n}]; wo1 = Table[{1, 2, 3, 4, 5}, {i1, 1, n}, {j1, 1 Total[Diagonal[ZX1[[zc1[[i1]], zc1 Do[zo1[[zc1[[i1]] [[Pos1[[j1]][ [1]]]], zc1 Style [MatrixForm[outputform1@Dis3[zwotype2[[j1]]]], RGBColor[0, 0.5, 0]]}, {j1, 1, Length[zwotype2]}]]; Str = "There are <> ToString[Sum[NumberofType3[zwotype2[[i1] Do[Do[Do[If[Min[zomatrix2[[i2,i2]] ∼ Join ∼ zomatrix2 [[j2, j2]]] > Max[zomatrix2[[i2,j2] Appendix IV: Some frequently appeared factors in mass relations…”

“…If one follows previous settings and approaches directly, the case n = 6 with general masses still seems far beyond reach due to the enormous amount of computations, and astounding complexity of polynomials involved. Finiteness for n = 6 has been proved for only a few exceptional cases [6,14,15,16,25], for some of which the exact number of central configurations were obtained.…”

Filaments and voids in planar central configurations

“…Central configurations (CC's) are a special class of configurations that give rise to the only known "explicit" solutions of the n-body problem. Regardless of its importance, only planar CC's (PCC's) of a low number of bodies have been computed [Moe89,Fer02,MZ19,DZD20], as well as spatial CC's of n = 5, 6 [MZ20] and n = 500 bodies of equal masses (SCCe's) [BGS03]. In this work, we have numerically computed PCC's of n = 1000 bodies of equal masses (PCCe's).…”