The restricted three-body problem and holomorphic curves
Every ``Pathways in Mathematics'' volume ``offers a roadmap to a currently well-developed mathematical research field''. The present book deals with the long history of the restricted three-body problem since its origin in a Birkhoff question from 1915. Its aim is perfectly resumed by the authors own words from Chapter 1, Introduction: ``The purpose of these notes is to make young ambitious researchers familiar with the main players of Birkhoff's question, namely the restricted three-body problem and disk-like global surfaces of section, and to introduce them to the modern techniques of symplectic geometry, which reduce Birkhoff's question to questions about symplectic embeddings and systolic geometry.'' Chapters 2 and 3 present symplectic geometry for Hamiltonian mechanics and symmetries. Chapter 4 is devoted to the regularization of two-body collisions. Chapters 5 and 6 deal with the restricted three-body problem and its relationship with contact geometry. Periodic orbits are presented for general Hamiltonians in Chapter 7 and for the problem studied in Chapter 8. Global surfaces of sections are the objects of Chapter 9 while their construction is postponed to Chapter 17. The other three chapters with general mathematical tools are: 10 for the Maslov index, 11 for the spectral flow and 12 for convexity. The next three chapters cover the finite energy planes while Chapter 16 discusses compactness. The last chapter offers a numerical point of view. The bibliography is rich and contains 245 titles. As conclusion, this very nice book gives an impressive picture of several recent tools from geometry and dynamics involved in a historical problem.
scientific article; zbMATH DE number 6928228