Martha Álvarez-Ramírez

arXiv2026

Nonlinear Stability, Resonances, and Singular Reduction in the Unequal-Mass Equilateral Restricted Four-Body Problem

We study the nonlinear stability of the elliptic equilibrium points $L_3$, $L_5$, and $L_6$ in the planar equilateral restricted four-body problem with unequal primary masses. We provide a systematic classification of nonlinear stability over the full two-parameter mass plane by combining high-order Birkhoff normal forms, numerical continuation, and singular reduction of resonant normal forms. In nonresonant regions, Arnold's theorem is applied when the quartic nondegeneracy condition holds, while sixth-order normalization resolves the degenerate cases. The $2$:$-1$ and $3$:$-1$ resonances are analyzed using the theorems of Alfriend, Markeev, and Meyer, including a classification of the $3$:$-1$ resonant dynamics. Singular reduction provides the corresponding reduced orbit spaces and reveals saddle-center bifurcations associated with the appearance and disappearance of periodic-orbit families. These results extend previous stability analyses beyond symmetric mass configurations and provide a geometric description of the resonant dynamics in the unequal-mass equilateral restricted four-body problem.

math.DS↗

arXiv2021

A note on the integrability of exceptional potentials via polynomial bi-homogeneous potentials

This paper is concerned with the polynomial integrability of the two-dimensional Hamiltonian systems associated to complex homogeneous polynomial potentials of degree $k$ of type $V_{k,l}=α(q_2-i q_1)^l (q_2+iq_1)^{k-l}$ with $α\in\mathbb{C}$ and $l=0,1,\dots, k$, called exceptional potentials. Hietarinta \cite{Hietarinta1983} proved that the potentials with $l=0,1,k-1,k$ and $l=k/2$ for $k$ even are polynomial integrable. We present an elementary proof of this fact in the context of the polynomial bi-homogeneous potentials, as was introduced by Combot et al. \cite{Combot2020}. In addition, we take advantage of the fact that we can exchange the exponents to derive an additional first integral for $V_{7,5}$, unknown so far. The paper concludes with a Galoisian analysis for $l=k/2$.

math.DS↗