Related papers: Central configurations, Morse and fixed point indi…

Central configurations of $n$ point particles in $E\approx \mathbb{R}^d$ with respect to a potential function $U$ are shown to be the same as the fixed points of the normalized gradient map $F=-\nabla_M U / \lVert \nabla_M U \rVert_M$,…

For the curved n-body problem, we show that the set of ordinary central configurations is away from most singular configurations in H^3, and away from a subset of singular configurations in S^3. We also show that each of the n!/2 geodesic…

Central configurations play an important role in the dynamics of the $n$-body problem: they occur as relative equilibria and as asymptotic configurations in colliding trajectories. We illustrate how they can be found as projective fixed…

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…

We consider the $N$-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of locked inertia tensor, we compute the moment of inertia for systems moving on spheres and hyperbolic spheres and show that…

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the…

Central configurations give rise to self-similar solutions to the Newtonian $N$-body problem, and play important roles in understanding its complicated dynamics. Even the simple question of whether or not there are finitely many planar…

We describe the configuration space $\mathbf{S}$ of polygons with prescribed edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on $\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are…

The classical theory of attractive central force problem on the standard (flat) Euclidean plane can be generalized to surfaces by reformulating the basic underlying physical principles by means of differential geometry. Attractive central…

We compute the Morse index $\textsf{m}(u_{p})$ of any radial solution $u_{p}$ of the semilinear problem: \begin{equation} \label{problemaAbstract}\tag{P} \left\{ \begin{array}{lr} -\Delta u=|x|^{\alpha}|u|^{p-1}u & \mbox{in } B\\ u=0 &…

A stochastic optimization algorithm for analyzing planar central and balanced configurations in the $n$-body problem is presented. We find a comprehensive list of equal mass central configurations satisfying the Morse equality up to $n=12$.…

Planar central configurations can be seen as critical points of the reduced potential or solutions of a system of equations. By the homogeneity and invariance of the potential with respect to SO(2), it is possible to see that the…

This is the second paper in our sequence. Here, we apply our abstract Morse index formulation developed in the previous paper to study several optimization set-ups with constraints, including type I or/and type II considerations. A common…

For the power-law potential $n$-body problem, we study a special kind of central configurations where all the masses lie on a circle and the center of mass coincides with the center of the circle. It is also called the centered co-circular…

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian $n$-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where $n$ particles of…

In the gravitation n-body Problem, a homothetic orbit is a special solution of the Newton's Equations of motion, in which each body moves along a straight line through the center of mass and forming at any time a central configuration. In…

In the Newtonian $n$-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for…

An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented…

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$}…

Let $Y\subseteq \mathbb{R}^n$ be a closed definable subset and $X\subseteq \mathbb{R}^n$ be a smooth manifold. We construct a version of Morse theory for the restriction to $X$ of the Euclidean distance function from $Y$. This is done using…