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

In this paper, as a continuation of [Fernandez-Guasti, $\mathit{\text{Celest}}\mathit{\text{Mech}}\mathit{\text{Dyn}}\mathit{\text{Astron}}$ 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 $\mathit{\text{particular}}\mathit{\text{integrals}}$ and the phenomenon of $\mathit{\text{particular}}\mathit{\text{involution}}$. 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.