Related papers: Balanced configurations of points in the plane
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every…
A plane configuration {v_1,...,v_m} of vectors in {\mathbb R}^2 is said to be balanced if for any index i, the set of the det(v_i,v_j) for j\neq i is symmetric around the origin. A plane configuration is said to be uniform if every pair of…
We show that the number of $\mathbf{S}$-balanced configurations of four bodies in the plane is finite, provided that the symmetric matrix $\mathbf{S}$ is close to a numerical matrix.
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…
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 equilibria of point particles under the action of two body central forces in which there are both repulsive and attractive interactions, often known as central configurations, with diverse applications in physics, in…
We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these…
Let $G$ be a group acting on a finite set $\Omega$. Then $G$ acts on $\Omega\times \Omega$ by its entry-wise action and its orbits form the basis relations of a coherent configuration (or shortly scheme). Our concern is to consider what…
We develop a systematic framework for constructing spherical harmonics on the two-dimensional unit sphere as superpositions of Gaussian beams whose poles form well-separated point configurations. The distributional and analytic properties…
For a dynamic system consisting of $n$ point vortices in an ideal plane fluid with a steady, incompressible and} irrotational background flow, a more physically significant definition of a fixed equilibrium configuration is suggested. Under…
We study equilibrium configurations of infinitely many identical particles on the real line or finitely many particles on the circle, such that the (repelling) force they exert on each other depends only on their distance. The main question…
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…
We present some methods for constructing connected spatial geometric configurations $(p_{q}, n_{k})$ of points and lines, preserved by the same rotations (and reflections) of Euclidean space $E^{3}$ as the chosen Platonic solid. In this…
In the Euclidean plane ${\bf{E}}^2$, fix four pairwise distinct points \begin{equation*} \label{eqA} \begin{array}{ccc} A=(a_1,a_2),\ B=(b_1,b_2),\ C=(c_1,c_2),\ D=(d_1,d_2), \end{array} \end{equation*} together with four non-zero real…
In this paper it is shown that every finite cyclic group satisfies the CI-property for the class of balanced configurations.
The balanced configurations are those n-body configurations which admit a relative equilibrium motion in a Euclidean space E of high enough dimension 2p. They are characterized by the commutation of two symmetric endomorphisms of the…
A central hyperplane arrangement in C^2 with multiplicity is called a `locus configuration' if it satisfies a series of `locus equations' on each hyperplane. Following Chalykh, Feigin and Veselov [CFV99], we demonstrate that the first locus…
We introduce ordered and unordered configuration spaces of 'clusters' of points in an Euclidean space $\mathbb{R}^d$, where points in each cluster satisfy a 'verticality' condition, depending on a decomposition $d=p+q$. We compute the…
We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m…
We review and develop the classical theory of moments of configurations of weighted points with a focus on systems with an identically vanishing first moment. The latter condition produces equations for equilibrium configurations of systems…