Symplectic Methods in the Numerical Search of Orbits in Real-Life Planetary Systems

SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3284-3319, December 2023.
Abstract. The intention of this article is to illustrate the use of methods from symplectic geometry for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g., the three-body problem). The main directions pursued in this article are as follows: (1) given two periodic orbits, decide when they can be connected by a regular family of periodic orbits; (2) use numerical invariants from Floer theory which help predict the existence of orbits in the presence of a bifurcation; (3) attach a sign [math] to each elliptic or hyperbolic Floquet multiplier of a closed symmetric orbit, which generalizes the classical Krein–Moser sign to also include the hyperbolic case; and (4) do all of the above in a visual, easily implementable, and resource-efficient way. The mathematical framework is provided by the first and third authors in [U. Frauenfelder and A. Moreno, J. Symplectic Geom., to appear], where, as it turns out, the “Broucke stability diagram” [R. Broucke, AIAA J., 7 (1969), pp. 1003–1009] was rediscovered, but further refined with the above signs and algebraically reformulated in terms of quotients of the symplectic group. The advantage of the framework is that it applies to the study of closed orbits of an arbitrary Hamiltonian system. We will carry out numerical work based on the cell-mapping method as described in [D. Koh, R. L. Anderson, and I. Bermejo-Moreno, J. Astronautical Sci., 68 (2021), pp. 172–196] for the Jupiter-Europa and Saturn-Enceladus systems. These are currently systems of interest, falling in the agenda of space agencies like NASA, as these icy moons are considered candidates for harboring conditions suitable for extraterrestrial life.