Morse Theory for S-balanced Configurations in the Newtonian n-body Problem

For the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the n-body is a (constant up to rotations and scalings) central configuration. For $$d\le 3$$ d ≤ 3 , the only possible homographic motions are those given by central configurations. For $$d \ge 4$$ d ≥ 4 instead, new possibilities arise due to the higher complexity of the orthogonal group $$\mathrm {O}(d)$$ O ( d ) , as observed by Albouy and Chenciner (Invent Math 131(1):151–184, 1998). For instance, in $$\mathbb {R}^4$$ R 4 it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for $$d\ge 4$$ d ≥ 4 there is a wider class of S-balanced configurations (containing the central ones) providing simple solutions of the n-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in $$\mathbb {R}^d$$ R d , for arbitrary $$d\ge 4$$ d ≥ 4 , and establish a version of the $$45^\circ $$ 45 ∘ -theorem for balanced configurations, thus answering some of the questions raised in Moeckel (Central configurations, 2014). Also, a careful study of the asymptotics of the coefficients of the Poincaré polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of S-balanced configurations. In the last part of the paper, we focus on the case $$d=4$$ d = 4 and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational n-body problem which improves a previous celebrated result of McCord (Ergodic Theory Dyn Syst 16:1059–1070, 1996).

On the Nature of Equilibrium Points in the Axisymmetric Five-Body Problem

The aim of this work is to numerically investigate the nature of the equilibrium points of the axisymmetric five-body problem. Specifically, we consider two cases regarding the convex or concave configuration of the four primary bodies. The specific configuration of the primaries depends on two angle parameters. Combining numerical methods with systematic and rigorous analysis, we reveal how the angle parameters affect not only the relative positions of the equilibrium points but also their linear stability. Our computations reveal that linearly stable equilibria exist in all possible central configurations of the primaries, thus improving and also correcting the findings of previous similar works.