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

The N-body problem of celestial mechanics, i.e. the motion of N heavenly bodies under the mutual gravitational attraction, has been a major driving force in the development of physics and mathematics since the times of Newton. For example, it led to the discovery of chaos as well as to the stability theory of Arnold, Kolmogorov and Moser. While the 2-body (or Kepler) problem is completely integrable and can be explicitly solved, already the case of 3 bodies exhibits intricate and sometimes chaotic dynamics that are far from understood. An interesting special case, proposed by Poincare, arises when the mass of the third body (the satellite) is negligible compared to the first two (the primaries). If in addition the two heavy bodies move on circles and all three bodies move in the same plane, then switching to rotating coordinates transforms the problem into the motion on a 3-dimensional energy level set in a time-independent Hamiltonian system with two degrees of freedom. This case is known as the (planar circular) restricted 3-body problem. Besides its theoretical interest, this problem is also of practical relevance for space orbit design because the motion of a satelite under the influence of earth and moon, or a spacecraft under the influence of sun and earth, roughly fit into this framework. The goal of this project is to study the dynamics of the restricted 3-body problem using techniques of modern symplectic topology, in particular the theory of holomorphic curves. More specific goals are the following:1. Producing finite energy foliations for the planar restricted 3-body problem below and above the first critical value.2. Studying families of periodic orbits in the spatial restricted 3-body problem and their omega-limit sets.3. Defining invariants to two-center Stark--Zeeman systems and applying them to the question of fibrewise convexity in the planar restricted 3-body problem.4. Establish properties of the Lagrange capacity, in particular its relation to Ekeland--Hofer capacities.5. Developing a mathematical theory of fuel minimization in space travel and explore its connections to notions in symplectic topology such as Floer homology and Mane's critical value.

• DFG Programme: Research Grants