We present a proof of the meromorphic non--integrability of the
planar $N$-Body Problem for some special cases. A simpler proof is
added to those already existing for the Three-Body Problem with
arbitrary masses. The $N$-Body Problem with equal masses is also
proven non-integrable. Furthermore, a new general result on
additional integrals is obtained which, applied to these specific
cases, proves the non-existence of an additional integral for the
general Three-Body Problem, and provides for an upper bound on the
amount of additional integrals for the equal-mass Problem for
$N=4,5,6$. These results appear to qualify differential Galois
theory, and especially a new incipient theory stemming from it, as
an amenable setting for the detection of obstructions to Hamiltonian
integrability.

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On the meromorphic non-integrability of some n-body problems

In this paper we study some necessary conditions and sufficient conditions for the nested polygonal solutions of planar 2N-body problems.

Periodic solutions for planar 2N-body problems

Central configurations of nested regular polyhedra for the spatial 2n-body problem

In this paper we prove the existence of central configurations of the pn-body problem where the masses are at the vertices of p nested regular n-gons with a common center for all p ≥ 2 and n ≥ 2. In such configurations all the masses on the same n-gon are equal, but masses on different n-gons could be different.

/pdf/on-the-existence-of-central-configurations-of-p-nested-n-3gvx5zjnwi.pdf

On the Existence of Central Configurations of p Nested n-gons

Double pyramidal central configurations

For n-body problems, a central configuration (CC) plays an im- portant role In this paper, we establish the relation between the spatial pyramidal central configuration (PCC) and the planar central configuration We prove that the base of PCC is also a CC and we also prove that for some given conditions a planar CC can be extended to a PCC In particular, if the pyramidal central configuration has a regular polygon base, then the masses of base are equal and the distance between the top vertex and the base is fixed and the mass of the top vertex is selective Furthermore, the pyramidal central configuration gives rise to an example of a perverse solution in R3

/pdf/pyramidal-central-configurations-and-perverse-solutions-98tb834w2u.pdf

Pyramidal central configurations and perverse solutions

A simpler proof of regular polygon solutions of the N-body problem☆

Nested regular polygon solutions of 2N-body problem☆

This paper we consider for the N -body problem with potential 1 /r α (0 < α < 1) the existence of hyperbolic motions for any prescribed limit shape and any given initial conﬁguration of the bodies. Here E is the Euclidean space where the bodies moving and k · k E is the norm induced by the inner product. The energy level h > 0 of the motion can also be chosen arbitrarily. We use the global viscosity solutions for the Hamilton-Jacobi equation H ( x, d x u ) = h and geodesics.

A N-body problem with weak force potential through Hamilton-Jacobi equation approach

In order to apply Morse's critical point theory, we use mutual distances as coordinates to discuss a kind of central configuration of the planar Newtonian 5-body problem with a trapezoidal convex hull which is convex but not strictly convex, i.e., four of the five bodies are located at the vertices of a trapezoid, and the fifth one is located on one of the parallel sides. We show that there is at most one central configuration of this geometrical shape for a given cyclic order of the five bodies along the convex hull. In addition, if the parallel side containing the three collinear bodies is strictly shorter than the other parallel side, the configuration must be symmetric, i.e., the trapezoid is isosceles, and the last body is at the midpoint of the shorter parallel side.

Shiqing Zhang

Papers

Pyramidal central configurations and perverse solutions

A simpler proof of regular polygon solutions of the N-body problem☆

Nested regular polygon solutions of 2N-body problem☆

A N-body problem with weak force potential through Hamilton-Jacobi equation approach

On the uniqueness of the planar 5-body central configuration with a trapezoidal convex hull