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Ernesto Pérez-Chavela

Publications and source records attributed to Ernesto Pérez-Chavela.

17 recordsLinked to original sources
arXiv2025

Equivalence of Rigid Motions and Relative Equilibria in the N-Body Problem on the Two-Sphere

We investigate the relationship between rigid motions and relative equilibria in the N-body problem on the two-dimensional sphere, S2. We prove that any rigid motion of the N-body system on S2 must be a relative equilibrium. Our approach extends the classical study of rigid body dynamics by Euler and utilizes a rotating frame attached to the particles to derive the corresponding equations of motion. We further show that our results can be extended to the N-body gravitational system in R3. The results are oriented to a broader understanding of the dynamics of N-body systems on curved surfaces.

math.DS↗
arXiv2024

Continuations and bifurcations of relative equilibria for the positive curved three body problem

The positive curved three body problem is a natural extension of the planar Newtonian three body problem to the sphere $\mathbb{S}^2$. In this paper we study the extensions of the Euler and Lagrange Relative equilibria ($RE$ in short) on the plane to the sphere. The $RE$ on $\mathbb{S}^2$ are not isolated in general. They usually have one-dimensional continuation in the three-dimensional shape space. We show that there are two types of bifurcations. One is the bifurcations between Lagrange $RE$ and Euler $RE$. Another one is between the different types of the shapes of Lagrange $RE$. We prove that bifurcations between equilateral and isosceles Lagrange $RE$ exist for equal masses case, and that bifurcations between isosceles and scalene Lagrange $RE$ exist for partial equal masses case.

math.CA↗
arXiv2024

Periodic orbits near collision in a restricted four-body problem for the eight choreography

We study orbits near collision in a non-autonomous restricted planar four-body problem. This restricted problem consists of a massless particle moving under the gravitational influence due to three bodies following the figure-eight choreography. We use regularized coordinates in order to deal numerically with motions near collision, and reversing symmetries to study both theoretically and numerically certain type of symmetric periodic orbits of the restricted system. The symmetric periodic orbits (initial conditions) were computed with the help of some boundary-value problems.

math.DS↗
arXiv2023

Three body relative equilibria on $\mathbb{S}^2$

We study relative equilibria ($RE$ in short) for three-body problem on $\mathbb{S}^2$, under the influence of a general potential which only depends on $\cosσ_{ij}$ where $σ_{ij}$ are the mutual angles among the masses. Explicit conditions for masses $m_k$ and $\cosσ_{ij}$ to form relative equilibrium are shown. Using the above conditions, we study the equal masses case under the cotangent potential. We show the existence of scalene and isosceles Euler $RE$, and isosceles and equilateral Lagrange $RE$.

math.CA↗
arXiv2023

A new method to study relative equilibria on $\mathbb{S}^2$

We develop a new geometrical technique to study relative equilibria for a system of $n$--positive masses, moving on the two dimensional sphere $\mathbb{S}^2$, under the influence of a general potential which only depends on the mutual distances among the masses. The big difficulty to study relative equilibria on $\mathbb{S}^2$, that we call $RE$ by short, is the absence of the center of mass as a first integral. We show that the two vanishing components of the angular momentum, for motions on $\mathbb{S}^2$, play the same role as the center of mass for motions on the Euclidean plane. From here we obtain that the rotation axis of a $RE$ is one of the principal axes of the inertia tensor. Conditions for have $RE$ and relations between the shape (given by the arc angles $σ_{ij}$ among the masses) and the configuration (given by the polar angles $θ_k$ and $ϕ_i - ϕ_j$ in spherical coordinates) are shown. For $n=3$, we show explicitly the conditions to have Euler and Lagrange $RE$ on $\mathbb{S}^2$. As an application of our method we study the the equal masses case for the positive curved three body problem where we show the existence of scalene and isosceles Euler $RE$ and isosceles Lagrange $RE$.

math.CA↗
arXiv2022

Equal masses Eulerian relative equilibria on a rotating meridian of S^2

Relative equilibria on a rotating meridian on $\mathbb{S}^2$ in equal-mass three-body problem under the cotangent potential are determined. We show the existence of scalene and isosceles relative equilibria. Almost all isosceles triangles, including equilateral, can form a relative equilibrium, except for the two equal arc angles $θ= π/2$. For $θ\in (0,2π/3)\setminus \{π/2\}$, the mid mass must be on the rotation axis, in our case, at the north or south pole of $\mathbb{S}^2$. For $θ\in (2π/3,π)$, the mid mass must be on the equator. For $θ=2π/3$, we obtain the equilateral triangle, where the position of the masses is arbitrary. When the largest arc angle $a_\ell$ is in $a_\ell\in (π/2,a_c)$, with $a_c=1.8124...$, two scalene configurations exist for given $a_\ell$.

math.CA↗
arXiv2021

Bifurcation of closed orbits from equilibria of Newtonian systems with Coriolis forces

We consider autonomous Newtonian systems with Coriolis forces in two and three dimensions and study the existence of branches of periodic orbits emanating from equilibria. We investigate both degenerate and nondegenerate situations. While Lyapunov's center theorem applies locally in the nondegenerate, nonresonant context, equivariant degree theory provides a global answer which is significant also in some degenerate cases. We apply our abstract results to a problem from Celestial Mechanics. More precisely, in the three-dimensional version of the Restricted Triangular Four Body Problem with possibly different primaries our results show the existence of at least seven branches of periodic orbits emanating from the stationary points.

math.DS↗
arXiv2020

Continuation of relative equilibria in the $n$--body problem to spaces of constant curvature

We prove that all non-degenerate relative equilibria of the planar Newtonian $n$--body problem can be continued to spaces of constant curvature $κ$, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. In particular, we extend Lagrange's triangle configuration with different masses to both positive and negative curvature spaces.

math.DS↗
arXiv2019

Regularization of the restricted $(n+1)$-body problem on curved spaces

We consider $(n+1)$ bodies moving under their mutual gravitational attraction in spaces with constant Gaussian curvature $κ$. In this system, $n$ primary bodies with equal masses form a relative equilibrium solution with a regular polygon configuration, and the remaining body of negligible mass does not affect the motion of the others. We show that the singularity due to binary collision between the negligible mass and the primaries can be regularized local and globally through suitable changes of coordinates (Levi-Civita and Birkhoff type transformations).

math.DS↗
arXiv2019

Relative equilibria for the positive curved $n-$body problem

We consider the $n$ body problem defined on surfaces of constant positive curvature. For the 5 and 7 body problem in a collinear symmetric configuration we obtain initial positions which lead to relative equilibria. We give explicitly the values of masses in terms of the initial positions. For positions for which relative equilibria exist, there are infinitely many values of the masses that generate such solutions. For the 5 and 7 body problem, the set of parameters (masses and positions) leading to relative equilibria has positive Lebesgue measure.

math.DS↗
arXiv2018

Symmetric Liapunov center theorem for orbit with nontrivial isotropy group

In this article we prove two versions of the Liapunov center theorem for symmetric potentials. We consider a~second order autonomous system $\ddot q(t)=-\nabla U(q(t))$ in the presence of symmetries of a compact Lie group $Γ$ acting linearly on $\mathbb{R}^n.$ We look for non-stationary periodic solutions of this system in a~neighborhood of an orbit of critical points of the potential $U.$

math.CA↗
arXiv2018

Symmetric Liapunov center theorem for minimal orbit

Using the techniques of equivariant bifurcation theory we prove the existence of non-stationary periodic solutions of $Γ$-symmetric systems $\ddot q(t)=-\nabla U(q(t))$ in any neighborhood of an isolated orbit of minima $Γ(q_0)$ of the potential $U$. We show the strength of our result by proving the existence of new families of periodic orbits in the Lennard-Jones two- and three-body problems and in the Schwarzschild three-body problem.

math.CA↗
arXiv2017

Stability interchanges in a curved Sitnikov problem

We consider a curved Sitnikov problem, in which an infinitesimal particle moves on a circle under the gravitational influence of two equal masses in Keplerian motion within a plane perpendicular to that circle. There are two equilibrium points, whose stability we are studying. We show that one of the equilibrium points undergoes stability interchanges as the semi-major axis of the Keplerian ellipses approaches the diameter of that circle. To derive this result, we first formulate and prove a general theorem on stability interchanges, and then we apply it to our model. The motivation for our model resides with the $n$-body problem in spaces of constant curvature.

math.DS↗
arXiv2016

Symmetric Liapunov center theorem

In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider the system $\ddot{q}= -\nabla U(q),$ where $U(q)$ is a $Γ$-symmetric potential, where $Γ$ is a compact Lie group acting linearly on $\mathbb{R}^n$. If the system possess a non-degenerate orbit of stationary solutions $Γ(q_0)$ with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian $\nabla^2 U(q_0)$, then in any neighbourhood of orbit $Γ(q_0)$ there is a periodic orbit of solutions of the system.

math.CA↗
arXiv2011

On polygonal relative equilibria in the N-vortex problem

Helmholtz's equations provide the motion of a system of N vortices which describes a planar incompressible fluid with zero viscosity. A relative equilibrium is a particular solution of these equations for which the distances between the vortices are invariant during the motion. In this article, we are interested in relative equilibria formed of concentric regular polygons of vortices. We show that in the case of one regular polygon (and a possible vortex at the center) with more than three vertices (two if there is a vortex at the center), a relative equilibrium requires equal vorticities (on the polygon). We also determine all the relative equilibria with two concentric regular n-gons and the same vorticity on each n-gon. This result completes the classical studies for two regular n-gons when all the vortices have the same vorticity or when the total vorticity vanishes.

math-ph↗