Trifolium Rose as an Analytic Four-Body Choreography

Abstract

The trajectories and interaction forces of four bodies with equal masses are analytically described. The bodies follow a choreographic motion in a trifolium rose periodic orbit without collisions. The center of mass is fixed at the origin. The nonzero angular momentum and the energy of the system are constant. The interaction forces are not Newtonian, but linear attractive and repulsive along the lines joining the bodies. Due to the linearity of the forces, the curve can be arbitrarily scaled in orthogonal directions. For equal scaling, several other quantities, such as the moment of inertia, as well as the kinetic and potential energies, are constant. This choreography is closely related to Pascal’s limaçon four-body choreography.

Keywords

Periodic orbit / Celestial choreographies / Plane curve trajectory / Centro-symmetric motion / 70F10 / 37N05 / 37C81 / 42A05

References

[1] Chenciner, A., Gerver, J., Montgomery, R., Simó, C.: Simple Choreographic Motions of N Bodies: a Preliminary Study. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-21791-6_9

[2] Chenciner, A., Venturelli, A.: Minima de l’intégrale d’action du problème Newtonien de 4 corps de masses égales dans R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R^3$$\end{document}: orbites ‘hip-hop’. Celest. Mech. Dyn. Astron. 77(2), 139–151 (2000). https://doi.org/10.1023/A:1008381001328

[3] Dana-Picard, T.: Automated study of a regular trifolium. Math. Comput. Sci. 13(1/2), 57–67 (2018). https://doi.org/10.1007/s11786-018-0351-7

[4] Fernández-Guasti, M.: The components exponential function in scator hypercomplex space: planetary elliptical motion and three body choreographies. In: Debnath, P., Srivastava, H.M., Chakraborty, K., Kumam, P. (eds) Advances in Number Theory and Applied Analysis, pp.195–230. World Scientific, Singapore (2023). https://doi.org/10.1142/13314

[5] Fernández-Guasti, M.: Analytic four-body limaçon choreography. Celest. Mech. Dyn. Astron. 137, 4 (2025). https://doi.org/10.1007/s10569-024-10235-x

[6] Kapela T, Simó C. Computer assisted proofs for nonsymmetric planar choreographies and for stability of the eight. Nonlinearity, 2007, 20(5): 1241-1255

[7] Lawrence JD. A Catalog of Special Plane Curves, 1972, New York, Dover Publications

Funding

Universidad Autonoma Metropolitana

RIGHTS & PERMISSIONS

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