Analytic Five-Body Gerono Lemniscata Choreography

Year 2025, Volume: 8 Issue: 4, 195 - 200

Manuel Fernandez-guasti

https://doi.org/10.33187/jmsm.1725741

Abstract

The trajectories and interaction forces of five bodies with equal masses are analytically described. The bodies follow a choreographic motion in a figure of eight, Gerono lemniscata periodic orbit without collisions. The forces are a linear combination of pairwise distances with either attractive or repulsive coefficients. The center of mass is fixed at the origin. The moment of inertia, as well as the kinetic and potential energies of the system, are constant. Notably, the angular momentum is zero just as in the Bernoulli Lemniscata parametrized by elliptic functions. The number of bodies in the choreography can be increased in a straightforward way. The orbit can be readily generalized to symmetric chains with an arbitrary number of loops.

Keywords

Analytical solutions , Celestial mechanics , n-body problem , Periodic orbits

Supporting Institution

The author acknowledges support by SECIHTI, project CF-2023-I-1864.

Project Number

CONAHCyT CF-2023-I-1864

References

• [1] C. Moore, Braids in classical dynamics, Phys. Rev. Lett., 70 (1993), 3675-3679. https://doi.org/10.1103/PhysRevLett.70.3675
• [2] A. Chenciner, R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152(3) (2000), 881-901.
• [3] C. Simo, New Families of Solutions in N-Body Problems, European Congress of Mathematics, (2001), 101-115.
• [4] T. Fujiwara, H. Fukuda, H. Ozaki, Choreographic three bodies on the lemniscate, J. Phys. A Math. Gen., 36 (2003), Article ID 2791. https://doi.org/10.1088/0305-4470/36/11/310
• [5] J. C. Lopez Vieyra, Five-body choreography on the algebraic lemniscate is a potential motion, Phys. Lett. A, 383(15) (2019), 1711-1715. https://doi.org/10.1016/j.physleta.2019.03.004
• [6] I. Popescu, L. Luca, S. S. Ghimisi. Mechanisms that generate Gerono’s lemniscate, IOP Conference Series: Materials Science and Engineering, 514 (2019), Article ID 012035. https://doi.org/10.1088/1757-899X/514/1/012035
• [7] M. Fernandez-Guasti, The components exponential function in scator hypercomplex space: Planetary elliptical motion and three body choreographies, In P. Debnath, H. M. Srivastava, K. Chakraborty, P. Kumam (Eds.), Advances in Number Theory and Applied Analysis, World Scientific, Singapore, 2023, pp. 195-230. https://doi.org/10.1142/13314
• [8] M. Fernandez-Guasti, Analytic four-body limac¸on choreography, Celestial Mech. Dynam. Astronom., 137(4) (2025), 1-12. https://doi.org/10.1007/s10569-024-10235-x
• [9] M. Fernandez-Guasti, Trifolium rose analytic four-body choreography, J. Appl. Math., 2025. Under review.
• [10] T. Fujiwara, H. Fukuda, H. Ozaki, N-body choreography on the lemniscate, Developments and Applications of Dynamical Systems Theory, 1369 (2004), 163-177.
• [11] A. Chenciner, J. Gerver, R. Montgomery, C. Sim´o, Simple Choreographic Motions of N Bodies: A Preliminary Study, In P. Newton, P. Holmes, A. Weinstein (Eds.), Geometry, Mechanics, and Dynamics, Springer-Verlag, NY, 2002. https://link.springer.com/chapter/10.1007/0-387-21791-6 9
• [12] G. Yu, Simple choreographies of the planar Newtonian N-Body problem, Arch. Ration. Mech. Anal., 225 (2017), 901-935. https://doi.org/10.1007/s00205-017-1116-1