Symplectic methods in the numerical search of orbits in real-life planetary systems

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: (1) given two periodic orbits, decide when they can be connected by a regular family; (2) use numerical invariants from Floer theory which help predict the existence of orbits in the presence of a bifurcation; (3) attach a sign +/- to each elliptic or hyperbolic Floquet multiplier of a closed symmetric orbit, generalizing 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, where the ``Broucke stability diagram'' was rediscovered, but further refined with the above signs, and algebraically reformulated in terms of GIT quotients of the symplectic group. The advantage of the above framework is that it applies to the study of closed orbits of an arbitrary Hamiltonian system. Moreover, in the case where the system admits symmetries in the form of ``reflections'', i.e. anti-symplectic involutions, which is the case for many systems of interest, the information provided for orbits which are symmetric is richer, and one may distinguish more symmetric orbits. This is the case for several well-known families in the space mission design industry, such as the Halo orbits, which are ubiquitous in real-life space missions. We will carry out numerical work based on the cell-mapping method, for the Jupiter-Europa and the 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 harbouring conditions suitable for extraterrestrial life.