arXiv subjects

Juan Manuel Sánchez-Cerritos

Publications and source records attributed to Juan Manuel Sánchez-Cerritos.

6 recordsLinked to original sources

arXiv2025

A Variational Approach to Planar Choreographies via Ekeland's Principle

We present a variational approach to obtain periodic solutions of the $N$-body problem, in particular the 'figure-eight' solution for three equal masses. The central idea is to explicitly optimize the \emph{spatial scale} within the Lagrangian action, leading to the functional $\mathcal F = K^{α/(α+2)} V^{2/(α+2)}$. We prove the existence of critical points of $\mathcal F$ that enforce a curve with a single self-crossing, and show that every reparametrized critical curve satisfies Newton's equations and is free of collisions. This framework recovers the Chenciner-Montgomery 'eight' (for $α=1$) and extends to the whole range $0<α<2$.

math.DS↗

arXiv2021

Eulerian collinear configuration for 3-body problem

For 3-body problem with any given masses $m_1, \,m_2,\,m_3>0$, there exist only Eulerian collinear central configuration and Lagrangian equilateral-triangle central configuration, and in this paper, for planar 3-body problem, we prove that there exists another non-collision trajectory $q$, which is not the variational minimizer of the Lagrangian action on the loop space $\overline{Λ_1}$, is also an Eulerian collinear central configuration at any instant. Moreover, we do not need the restriction condition on the winding number $deg(q_i-q_j)\neq0 \,(i\neq j)$.

math.DS↗

arXiv2020

Notes on spatial twisted central configurations for $2N$-body problem

We study the spatial central configuration formed by two twisted regular $N$-polygons. For any twist angle $θ$ and any ratio of the masses $b$ in the two regular $N$-polygons, we prove that the sizes of the two regular $N$-polygons must be equal.

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↗

arXiv2018

Choreography solutions of the $n$-body problem on $S^2$

We try to prove the existence of choreography solutions for the $n-$body problem on $S^2$. For the three-body problem, we show the existence of the 8-shape orbit on $S^2$.

math.DS↗

arXiv2017

Stability of fixed points and associated relative equilibria of the $3$-body problem on $\mathbb S^1$ and $\mathbb S^2$

We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to $\mathbb S^1$, but unstable if the bodies are considered in $\mathbb S^2$.

math.DS↗