Symplectic techniques in the restricted three body problem

• Applicants: Professor Dr. Kai Cieliebak; Professor Dr. Urs Frauenfelder
• Subject Area: Mathematics
• Term: from 2016 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 316136360

• Final Report Year: 2024

Final Report Abstract

The goal of this project was to use techniques of modern symplectic topology in order to study the dynamics of the restricted 3-body problem of celestial mechanics. Related to this goal, the following results were obtained: ˆrealization of several limiting cases of the restricted 3-body problem as concave toric domains;ˆ definition of J+ -like invariants for two-center Stark-Zeeman systems and investigation of their properties;ˆ computation of the Lagrangian capacity for concave and convex toric domains and proof of a conjectural relation to the Ekeland-Hofer capacities; ˆ progress on the dynamics of the restricted 3-body problem below and above the Mane critical value;ˆ development of symplectic techniques to aid numerical computations in space mission design; ˆ introduction and computation of versions of Rabinowitz Floer homology for Stark-Zeeman systems, with applications to existence of special types of orbits;ˆ analytical proofs for the existence of several period orbits that play an important role in the semiclassical theory of the helium atom. Besides solving several open problems, these results also open up some promising new directions of research in pure mathematics, theoretical physics, and engineering.

Publications

• Computing the ECH capacities of the rotating Kepler problem, PhD thesis, University of Augsburg (2020)
  A. Mohebbi
  
• Riemannian geometry of groups of diffeomorphisms preserving a stable Hamiltonian structure, PhD thesis, University of Augsburg (2020)
  K. Helmsauer
  
• Second species orbits of negative action and contact forms in the circular restricted three-body problem PhD thesis, University of Augsburg (2021)
  R. Nicholls
  
• The Regularized Free Fall I. Index Computations. Russian Journal of Mathematical Physics, 28(4), 464-487.
  Frauenfelder, U. & Weber, J.
  
• + -invariants for planar two-center Stark–Zeeman systems. Ergodic Theory and Dynamical Systems, 43(7), 2258-2292.
  Cieliebak, Kai; Frauenfelder, Urs & Zhao, Lei
  
• A variational approach to frozen planet orbits in helium. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 40(2), 379-455.
  Cieliebak, Kai; Frauenfelder, Urs & Volkov, Evgeny
  
• Cube normalized symplectic capacities
  J. Gutt, M. Pereira & V. Ramos
  
• Equivariant symplectic homology, linearized contact homology and the Lagrangian capacity, PhD thesis, University of Augsburg (2022)
  M. Pereira
  
• First Steps in Twisted Rabinowitz Floer Homology PhD thesis, University of Augsburg (2022)
  Y. Bähni
  
• J + -like Invariants under Bifurcations
  A. Mai
  
• On Langmuir’s periodic orbit. Archiv der Mathematik, 118(4), 413-425.
  Cieliebak, K.; Frauenfelder, U. & Schwingenheuer, M.
  
• On the Lagrangian capacity of convex or concave toric domain
  M. Pereira
  
• On the problem of convexity for the restricted three-body problem around the heavy primary. Hokkaido Mathematical Journal, 51(2).
  Frauenfelder, Urs; van Koert, Otto & Zhao, Lei
  
• The Stark problem as a concave toric domain. Geometriae Dedicata, 217(1).
  Frauenfelder, Urs
  
• First steps in twisted Rabinowitz–Floer homology. Journal of Symplectic Geometry, 21(1), 111-158.
  Bähni, Yannis
  
• Lagrangian Rabinowitz Floer Homology and its Application to Powered Flyby Orbits in the Restricted Three Body Problem, PhD thesis, University of Augsburg (2023)
  K. Ruck
  
• Nondegeneracy and integral count of frozen planet orbits in helium. Tunisian Journal of Mathematics, 5(4), 713-770.
  Cieliebak, Kai; Frauenfelder, Urs & Volkov, Evgeny
  
• On doubly symmetric periodic orbits. Celestial Mechanics and Dynamical Astronomy, 135(2).
  Frauenfelder, Urs & Moreno, Agustin
  
• On GIT quotients of the symplectic group, stability and bifurcations of periodic orbits (with a view towards practical applications). Journal of Symplectic Geometry, 21(4), 723-773.
  Frauenfelder, Urs & Moreno, Agustin
  
• Pseudo-Holomorphic Hamiltonian Systems
  F. Wagner
  
• Symplectic Methods in the Numerical Search of Orbits in Real-Life Planetary Systems. SIAM Journal on Applied Dynamical Systems, 22(4), 3284-3319.
  Frauenfelder, Urs; Koh, Dayung & Moreno, Agustin
  
• Generalization of Arnold’s J+-invariant for pairs of immersions. Journal of Topology and Analysis, 1-35.
  Häußler, Hanna