On the four-body limaçon choreography: maximal superintegrability and choreographic fragmentation

2025/04/23 by A. M. Escobar-Ruiz, Escobar-Ruiz, Adrian M, M. Fernández‐Guasti +1
Physics and Astronomy · #Black Holes and Theoretical Physics #Continuation #FOS: Physical sciences #Hamiltonian (control theory) #Mathematical Physics (math-ph) #Motion (physics) #Planar #Quadratic equation #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Space (punctuation)

paper · pdf · doi:10.48550/arxiv.2504.16403

openalex publication_date 2025/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, as a continuation of [Fernandez-Guasti, Celest Mech Dyn Astron 137, 4 (2025)], we demonstrate the maximal superintegrability of the reduced Hamiltonian, which governs the four-body choreographic planar motion along the trisectrix limaçon (resembling a folded figure eight), in the six-dimensional space of relative motion. The pairwise interaction potential V(r_ij) among the four bodies is a quadratic expression in the relative distances r_ij, with a combination of positive and negative coefficients. The corresponding eleven integrals of motion in the Liouville-Arnold sense are presented explicitly. Specifically, it is shown that the reduced Hamiltonian admits complete separation of variables in Jacobi-like variables. The emergence of this choreography is not a direct consequence of maximal superintegrability. Rather, it originates from the existence of particular integrals and the phenomenon of particular involution. We also provide a detailed analysis of the fragmentation of a general four-body choreographic motion into two isomorphic two-body choreographies, as well as the reverse process, namely, the fusion of two-body choreographies into a four-body configuration. This model combines choreographic motion with maximal superintegrability, a seldom-studied interplay in classical mechanics.

Citations

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• Superintegrability in classical mechanics
• Construction of new integrable Hamiltonians in two degrees of freedom
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Related

• N-body choreographies on a p-limacon curve
• The Lagrange reduction of the N-body problem, a survey
• A restricted four-body problem for the eight figure choreography
• Decomposition of the Hessian matrix for action at choreographic three-body solutions with figure-eight symmetry
• A Variational Approach to Planar Choreographies via Ekeland's Principle