Non-integrability and dynamics on the collision manifold of the pyramidal N+1 body problem
(角錐N＋１体問題の非可積分性と衝突多様体上のダイナミクス）

We study a special case of the N-body problem called the pyramidal
N + 1 body problem. We prove non-integrability of the pyramidal N + 1 body
problem, and we also prove non-existence of heteroclinic orbits on the
collision manifold for N larger than 472. First, we write down
differential equations of the pyramidal N +1 body problem. That is
represented by a Hamiltonian system with homogeneous potential of
degree －1. Hence, if we assume that λ is an eigenvalue of Hessian
matrix of potential U at solution of r = grad U(r) and there exist λ
such that square root of 9 － 8λ is not an odd number, then that
Hamiltonian system is non-integrable. Using this, we prove
non-integrability of the pyramidal N + 1 body problem. Using
McGehee’s transformation, we derive the total collision manifold of
the pyramidal N + 1 body problem, and we restrict the phase space to
the manifold. The pyramidal N + 1 body problem is non-integrable, but
taking some limit for mass ratio, we can obtain two integrable
systems. After introducing local coordinates to the collision manifold
for N larger than 472, we take some curves correspond to the orbits of
such integrable systems. By checking some inequarities on those
curves, we prove non-existence of heteroclinic orbits on the collision
manifold for N larger than 472.