arXiv subjects

Eduardo S. G. Leandro

Publications and source records attributed to Eduardo S. G. Leandro.

3 recordsLinked to original sources

arXiv2026

Moments, Equilibrium Equations and Mutual Distances

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 of interacting particles under the sole condition that interactions are between pairs of particles and along the lines connecting such pairs. Complying external forces are admitted, so the description of some dynamical equilibrium configurations, such as relative equilibria in Celestial Mechanics, is included in our approach. Moments provide a unified framework for equilibrium problems in arbitrary dimensions. The equilibrium equations are homogeneous and invariant by isometries (for interactions depending only on mutual distances), and are obtained through simple algebraic procedures requiring neither reduction by isometries nor a variational principle for their determination. Our equations include the renowned set of $n$-body central configuration equations by A. Albouy and A. Chenciner. These equations are extended to a rather broad class of equilibrium problems, and new equilibrium equations written in terms of mutual distances are introduced. We also apply moments to the theory of constraints for mutual distances of configurations of fixed dimension, and for co-spherical configurations, thus re-obtaining and adding to classical results by A. Cayley and successors. For the sake of concreteness, novel sets of central configurations equations are provided.

math-ph↗

arXiv2018

Schrödinger formalism for a particle constrained to a surface in $\mathbb{R}_1^3$

In this work it is studied the Schrödinger equation for a non-relativistic particle restricted to move on a surface $S$ in a three-dimensional Minkowskian medium $\mathbb{R}_1^3$, i.e., the space $\mathbb{R}^3$ equipped with the metric $\text{diag}(-1,1,1)$. After establishing the consistency of the interpretative postulates for the new Schrödinger equation, namely the conservation of probability and the hermiticity of the new Hamiltonian built out of the Laplacian in $\mathbb{R}_1^3$, we investigate the confining potential formalism in the new effective geometry. Like in the well-known Euclidean case, it is found a geometry-induced potential acting on the dynamics $V_S = - \frac{\hbar^{2}}{2m} \left(\varepsilon H^2-K\right)$ which, besides the usual dependence on the mean ($H$) and Gaussian ($K$) curvatures of the surface, has the remarkable feature of a dependence on the signature of the induced metric of the surface: $\varepsilon= +1$ if the signature is $(-,+)$, and $\varepsilon=1$ if the signature is $(+,+)$. Applications to surfaces of revolution in $\mathbb{R}^3_1$ are examined, and we provide examples where the Schrödinger equation is exactly solvable. It is hoped that our formalism will prove useful in the modeling of novel materials such as hyperbolic metamaterials, which are characterized by a hyperbolic dispersion relation, in contrast to the usual spherical (elliptic) dispersion typically found in conventional materials.

math-ph↗

arXiv2017

Factorization of the Stability Polynomials of Ring Systems

Let $D_n$ be the dihedral group with $2n$ elements, and suppose $n$ is greater than one. We call ring system a finite $D_n$-symmetric set of points in $\mathbb{R}^2$. Ring systems have been used as models for planets surrounded by rings, and may be seen as relative equilibria of the $N$-body or the $N$-vortex problem. As a first significant step towards linear stability analysis, we study the factorization of the stability polynomial of an arbitrary ring system by systematically exploiting the ring's symmetry through representation theory of finite groups. Our results generalize contributions by J. C. Maxwell from mid-XIX century until contemporary authors such as J. Palmore and R. Moeckel, among others.

math.DS↗