A Stellar Afternoon in Symplectic Dynamics

Cengiz Aydin is currently a postdoctoral fellow in the symplectic group at Heidelberg University, Germany. In 2023, he completed his PhD at the Université de Neuchâtel, Switzerland, under the supervision of Prof. Felix Schlenk, with co-supervision by Prof. Urs Frauenfelder. His doctoral research focused on symplectic geometry and its applications in celestial mechanics, particularly the Hill three-body problem. Prior to his PhD, he earned a Master's degree from the University of Augsburg in 2019, under the guidance of Prof. Urs Frauenfelder. At Heidelberg, he continues to explore the connections between symplectic geometry and celestial mechanics, conducting both theoretical and numerical studies.

In this talk I discuss applications of symplectic invariants to the Hill problem. The symplectic tools include computation of Conley-Zehnder index, Krein signature, and local Floer homology (graded by Conley-Zehnder indices) and its Euler characteristics. Significantly, the latter is a bifurcation-invariant. The first application I show is how to express lunar months in terms of Conley-Zehnder indices and Floquet multipliers associated to Hill's lunar orbit. As next and extensive application I demonstrate how such symplectic invariants help to analyze the interconnectedness and common network between natural families of symmetric periodic orbits. Most results that I present are based on a current work with Alexander Batkhin.

Arnaud Boutonnet is an aerospace engineer who works at the ESA European Space Operations Centre. He is mission analyst within the flight dynamics division. Arnaud received a master's degree in Electrical Engineering from the Institut Superieur d'Electronique de Paris, a post-graduate degree in Astrophysics from SUPAERO and a doctoral degree in Aerospace Engineering also from SUPAERO. His research interests include optimal control and trajectory optimization. Additional interests include spacecraft navigation, planetary protection and close proximity operations. As a senior staff, Arnaud is the mission analysis lead for JUICE. He also supports or has supported other missions, for instance Clear Space 1 (Active Debris Removal) and Neosat (electrical orbit raising). Finally Arnaud co-chairs the CNES Comet-Orb Centre of Expertise on astrodynamics.

In April 2023, ESA launched a large-class mission to Jupiter: JUICE, which stands for Jupiter Icy Moons Explorer. The mission currently relies on a series of 35 gravity assists with the Galilean moons Europa, Ganymede and Callisto, which is followed by an insertion into orbit around Ganymede. The mission is subject to several objectives (scientifically-driven, Delta-V, and time-of-flight) and constraints (again scientific, but also operational and platform related like power or ionisation dose). The design of the tour relies on a series of specific techniques - or cooking recipes - like full or pseudo-resonances, pi-transfer (a.k.a. backflip), broken-plane resonant transfer, Delta-V resonant transfer, low energy endgame, ... All these techniques rely on special properties of the two-, three- and four-body dynamics. The presentation will give an overview of the mission, the cooking recipes and their application leading to the current trajectory.

Born December 18, 1974 in Switzerland
2003: PhD at ETH Zurich under the supervision of Dietmar Salamon
2009: Habilitation at Ludwig-Maximilian University in Munich
2009-2014: Assistant and Associate Professor at Seoul National University in South Korea Since 2014: Professor at Augsburg University

Research interests: I am working in Symplectic Geometry, especially Floer homology, and its application to Hamiltonian dynamics. In particular, I am interested how the modern tools of Symplectic and Contact Geometry can be applied to old problems in celestial mechanics with a special focus on the restricted three-body problem.

In the study of families of periodic orbits it is important to distinguish left from right. Retrograde periodic orbits have a completely different behaviour than direct orbits. However, the spectrum of a symplectic matrix cannot distinguish left from right. In joint work with Agustin Moreno we showed how to associate to a symmetric periodic orbit two B-signs. These B-signs lead to an enhancement of the classical Broucke diagram. For elliptic eigenvalues they correspond to the Krein sign. However, they cannot just distinguish left from right for elliptic eigenvalues, but even lead to a notion of left and right for hyperbolic eigenvalues. These B-signs play as well an important role to understand bifurcations of families of periodic orbits and in particular about the behaviour of Maslov indices at bifurcations. These Maslov indices for families are crucial in the recent global approach to networks of periodic orbits using bifurcation graphs due to Aydin and Bathkin.

Jim Green has worked at NASA for 42 years before retiring in December 2022. He received his Ph.D. in Physics from the University of Iowa in 1979 and worked at Marshall Space Flight Center, Goddard Space Flight Center, and NASA Headquarters. During Jim's long career at NASA, he has been NASA's Chief Scientist and was the longest serving director of NASA's Planetary Science Division with the overall programmatic responsibility for the New Horizons spacecraft flyby of Pluto, the Juno spacecraft to Jupiter, and the landing of the Curiosity rover on Mars, just to name a few. Jim has received the Exceptional Achievement Medal for the New Horizons flyby of the Pluto system and NASA's highest honor, the Distinguish Service Medal. He has written over 125 scientific articles in refereed journals and over 80 technical and popular articles. In 2015, Jim coordinated NASA's involvement with the film The Martian. In 2017 Asteroid 25913 was renamed Jamesgreen in his honor.

Based on the analysis of the Apollo lunar samples, scientists believe that the Moon was formed out of a collision between the Earth with a Mars sized planet named Theia at a very early stage of the development of the solar system (~4.5 Ga years ago). From then on, the Earth and the Moon's evolution have been intertwined. The Moon has kept the Earth's rotational axis pointing in the same direction providing a significant level of stability for the Earth's climate. Very large impacts occurred early in lunar history followed by a time of enormous volcanic activity (~4.2-3.16 Ga) and this is believed to be a record of giant planet migration in the outer part of our solar system. Today, the Moon holds many fascinating mysteries for scientists to explore. Scientifically there has been several stunning advances in lunar science and a realization that going back to the Moon will provide scientists with the opportunity to accomplish transformational science in understanding the origin and evolution of our solar system. NASA's future plan is to go to the Moon to stay and then onto Mars.

Dr. Bhanu Kumar is currently a postdoctoral researcher in the Geometry and Dynamics group of the Institute for Mathematics, Universität Heidelberg, which he joined in May 2024. He is currently on leave from his previous position as a US National Science Foundation Postdoctoral Research Fellow, during which he was in the Mission Design and Navigation Section of the Jet Propulsion Laboratory, California Institute of Technology. He holds a Ph.D. in Mathematics from Georgia Tech, advised by Prof. Rafael de la Llave, as well as a Masters in Aerospace Engineering. His research interests focus on analytical and computational methods for and application of dynamical structures, especially around mean motion resonances, to design low and zero-fuel trajectories for missions in various planetary systems.

Understanding the dynamical structure of cislunar space beyond geosynchronous orbit is of significant importance for lunar exploration, as well as for design of high Earth-orbiting mission trajectories in other contexts. A key aspect of these dynamics is the presence of mean motion resonances. A number of prior spacecraft have been placed into stable lunar resonant orbits, such as NASA's TESS and IBEX, while some seemingly orbit in the unstable resonance regime (e.g. Russia's Spektr-R). Although libration point orbits have been studied extensively in the Earth-Moon system, the intricate nature of mean motion resonances, especially their stable and unstable orbit families and their overlapping heteroclinic connections, is less explored. The presence of such paths can be used beneficially for trajectory design, but could be potentially hazardous if unanticipated in mission planning.
While heteroclinic connections between resonances are fundamental to changing and understanding the evolution of a spacecraft's semimajor axis, they remain woefully understudied in the Earth-Moon system. In this talk, first we compute and analyze several important resonant orbit families within the Earth-Moon system based on the planar circular restricted 3-body problem. Focusing on interior resonances 4:1, 3:1, and 2:1, we identify a number of bifurcations of these resonances' periodic orbit families. Once the aforementioned orbits are computed, we then compute their stable and unstable manifolds using a parameterization method combined with a Poincaré map at osculating orbit perigee. Carrying out these computations across a range of Jacobi constants, we are able to characterize the range of naturally attainable semimajor axis values for future distant Earth-bound or lunar spacecraft missions.

Rita Mastroianni is a research fellow of the ACT (Advanced Concepts Team) at ESA/ESTEC. She has a master's degree in Pure and Applied Mathematics from the University of Rome "Tor Vergata", as well as a Ph.D. in Mathematics, from the University of Padova, under the supervision of Prof. Christos Efthymiopoulos and co-supervision of Prof. Ugo Locatelli. The focus of her doctoral research was on methods stemming from the use of canonical perturbation theory in a class of nearly-integrable problems arising from models of 3D planetary systems, especially for the study of exoplanetary dynamics. To this end, her research focused on a comparison between extended numerical simulations and the outcome of (semi-)analytical models constructed within the framework of canonical perturbation theory, as well as the Kolmogorov normal form in KAM theory. Her scientific interests focus on the applications of Hamiltonian and canonical perturbation theory in celestial mechanics.

In [1] we revisit the problem of understanding the structure of the phase space of the secular 3D planetary three body problem, focusing on the transition, as the level of mutual inclination increases, from a 'planar-like' to the 'Lidov-Kozai' regime. The latter is dominated by two different families of nearly circular and highly inclined periodic orbits, of which one becomes unstable via the usual Lidov-Kozai mechanism. Using a typical 'non-hierarchical' example, compatible with the exoplanetary ν-Andromedæ system, we show how the latter families are connected to the apsidal corotation families via a chain of saddle-node bifurcations. Moreover, in [2], we analyze, through a geometric description, the sequence of bifurcations of periodic orbits in an integrable Hamiltonian model derived from a normalization of the secular 3D planetary three body problem presented. Stemming from the results in [1], we analyze the phase space of the corresponding integrable Hamiltonian model. In particular, we propose a normal form leading to a integrable Hamiltonian whose sequence of bifurcations is qualitatively the same as the one in the complete system. Using a representation of the phase space in the 3D-sphere through Hopf variables ( [3], [4], [5]), we geometrically analyze the sequence of bifurcations leading to the appearance of new fixed points of the secular Hamiltonian, i.e., periodic orbits in the complete system. Finally, through a semi-analytical method, we find the critical values of the integral giving rise to the pitchfork and saddle-node bifurcations present in the system.

[1] R. Mastroianni, C. Efthymiopoulos. The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination, Celestial Mechanics and Dynamical Astronomy, 135(3):22, (2023);
[2] R. Mastroianni, A. Marchesiello, C. Efthymiopoulos, G. Pucacco. Bifurcations of periodic orbits in the 3D secular planetary 3-Body problem: an approach through an integrable Hamiltonian system, arXiv preprint (2024);
[3] M. Kummer. On resonant non linearly coupled oscillators with two equal frequencies, Commu- nications in Mathematical Physics, 48:53-79, (1976);
[4] R.H. Cushman and L.M. Bates. Global aspects of classical integrable systems, Birkhauser, (1997);
[5] A. Marchesiello and G. Pucacco. Bifurcation sequences in the symmetric 1:1 Hamiltonian res- onance, International Journal of Bifurcation and Chaos, 26(04): 1630011 (2016)

Agustin Moreno is a Junior Professor of Mathematics at Heidelberg University, and a former Member of the School of Mathematics at the Institute for Advanced Study, Princeton. He holds degrees from the University of Cambridge, and from the Humboldt University in Berlin, where he finished his Ph.D. He has been affiliated with the Berlin Mathematical School, and has been a Research Fellow at the Mittag-Leffler Institute in Sweden, where much of his work on the three-body problem was carried out. His research lies at the crossroads of geometry, topology, dynamics and mathematical physics, with further interest in applications to space mission design and astrodynamics, which he pursues in collaboration with aerospace and navigational engineers at NASA, and the group of Prof. Scheeres at the University of Colorado Boulder.

In this talk, I will briefly describe some of the theoretical aspects of the circular restricted three-body problem, from the point of view of symplectic geometry, and summarise some of the results obtained by our symplectic working group at the University of Heidelberg.