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

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 $$V(r_{ij})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>ij</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> among the four bodies is a quadratic expression in the relative distances $$r_{ij}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>ij</mml:mi> </mml:mrow> </mml:msub> </mml:math> , 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.