Total Collisions in the N-Body Shape Space

We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.

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).

Comparison between Vortex-In-Cell and Vortex Particle Methods on Two-Dimensional Problem

This research is concerned with the two-dimensional vortex method (VM) solvers. We develop and investigate the performance of the Vortex-In-Cell (VIC) and Vortex Particle Method (VPM) which are well known as the VM’s family members. The advantage of these both methods are that we can accelerate velocity computation procedure, an N-body problem in numerical methods, by using Fast Fourier Transform (FFT) and Fast Multipole Method (FMM), respectively. In addition, the viscous calculation process in VPM can be accelerated by using a scheme of Nearest Neighbor Particle Searching (NNPS) algorithms. Moreover, the no-through boundary condition treatment issue can be easily handled by using an immersed boundary condition for both methods. The accuracy and numerical cost of both numerical methods will be examined by simulating flow over an Impulsively Started Circular Cylinder and comparisons