Symplectic Geometry and N-body Problem

• Applicant: Privatdozent Dr. Lei Zhao
• Subject Area: Mathematics
• Term: from 2017 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 359698948

• Final Report Year: 2025

Final Report Abstract

In this project we have proposed to conduct research on interactions of symplectic geometry and the Newtonian N-body problem. We have made significant progress along various research lines. We have used tools from symplectic topology to understand the existence of a global surface of section in the planar circular restricted three-body problem and re-established this classical result around the heavy body. We have used Rabinowitz-Floer theory to show the existence and abundance of consecutive collisional orbits in restricted planar circular three-body problems. We have showed the existence of infinitely many generalized periodic orbits in a forced Kepler problem in an Euclidean space of arbitrary dimension, covering many important cases of restricted problems encounter in celestial mechanics. as a successful combination of Floer-theoretical techniques with localization and regularization considerations. We developed theory of J + -type invariants for two-center Stark-Zeeman systems. We obtained a simple shooting argument for the existence of frozen planet orbits in a Helium model. We developed the theory of projective and conformal corresponding natural mechanical systems. This theory was then used to establish the integrability of various types of natural mechanical billiards. We have developed several algorithms combining stochastic methods and rigorous numerics to improve the list of known planar central configurations in the equal mass case for up to 14 bodies, and we have made a first list of planar balanced configurations for up to 5 bodies in many cases. We have shown that the number of balanced configurations of 4 bodies in the plane with respect to a symmetric matrix that are sufficiently close to central configurations are finite up to similitudes.

Publications

• Existence of either a periodic collisional orbit or infinitely many consecutive collision orbits in the planar circular restricted three-body problem. Mathematische Zeitschrift, 291(1-2), 215-225.
  Frauenfelder, Urs & Zhao, Lei
  
• Lambert's theorem and projective dynamics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 377(2158), 20180417.
  Albouy, Alain & Zhao, Lei
  
• Periodic solutions and regularization of a Kepler problem with time-dependent perturbation. Transactions of the American Mathematical Society, 372(1), 677-703.
  Boscaggin, Alberto; Ortega, Rafael & Zhao, Lei
  
• Transverse Regularizations of Central Force Problems by Hamiltonian Structure, Proceedings of ICCM 2018
  U. Frauenfelder & L. Zhao
  
• Lambert’s Theorem on the Sphere, Rendiconti del Seminario Mathematico (Turin)
  L. Zhao
  
• Partial coherent state transforms, G × T-invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groups. Advances in Mathematics, 368, 107139.
  Mourão, José M.; Nunes, João P. & Pereira, Miguel B.
  
• Generalized periodic orbits in some restricted three-body problems. Zeitschrift für angewandte Mathematik und Physik, 72(1).
  Ortega, Rafael & Zhao, Lei
  
• Generalized periodic orbits of the time-periodically forced Kepler problem accumulating at the center and of circular and elliptic restricted three-body problems. Mathematische Annalen, 385(1-2), 59-99.
  Zhao, Lei
  
• Projective dynamics and an integrable Boltzmann billiard model. Communications in Contemporary Mathematics, 24(10).
  Zhao, Lei
  
• A numerical analysis of planar central and balanced configurations in the (n+1)-body problem with a small mass
  A. Doicu; A. Doicu & L. Zhao
  
• A stochastic optimization algorithm for analyzing planar central and balanced configurations in the n-body problem. Celestial Mechanics and Dynamical Astronomy, 134(3).
  Doicu, Alexandru; Zhao, Lei & Doicu, Adrian
  
• Darboux Inversions of the Kepler Problem. Regular and Chaotic Dynamics, 27(3), 253-280.
  Albouy, Alain & Zhao, Lei
  
• J+-invariants for planar two-center Stark–Zeeman systems. Ergodic Theory and Dynamical Systems, 43(7), 2258-2292.
  Cieliebak, Kai; Frauenfelder, Urs & Zhao, Lei
  
• On the problem of convexity for the restricted three-body problem around the heavy primary. Hokkaido Mathematical Journal, 51(2).
  FRAUENFELDER, Urs; KOERT, Otto van & ZHAO, Lei
  
• Boltzmann’s Billiard Systems: Computation of the Billiard Mapping and Some Numerical Results
  M. Plum; A. Takeuchi & L. Zhao
  
• Projective integrable mechanical billiards. Nonlinearity, 37(1), 015011.
  Takeuchi, Airi & Zhao, Lei
  
• Shooting for collinear periodic orbits in the Helium model. Zeitschrift für angewandte Mathematik und Physik, 74(6).
  Zhao, Lei
  
• Conformal transformations and integrable mechanical billiards. Advances in Mathematics, 436, 109411.
  Takeuchi, Airi & Zhao, Lei
  
• Geometric properties of integrable Kepler and Hooke billiards with conic section boundaries. Journal of Geometry and Physics, 204, 105289.
  Jaud, Daniel & Zhao, Lei
  
• Integrable Mechanical Billiards in Higher-Dimensional Space Forms. Regular and Chaotic Dynamics, 29(3), 405-434.
  Takeuchi, Airi & Zhao, Lei
  
• On the finiteness issue of four-body balanced configurations in the plane
  Y. Wang & L. Zhao
  
• Finiteness of non-degenerate central configurations of the planar n-body problem with a homogeneous potential
  J. Natrup; Q. Wang & Y. Wang