Intro; Contents; Chapter 1 Introduction; 1.1 The Birkhoff conjecture; 1.2 The power of holomorphic curves; 1.3 Systolic inequalities and symplectic embeddings; 1.4 Beyond the Birkhoff conjecture; Chapter 2 Symplectic Geometry and Hamiltonian Mechanics; 2.1 Symplectic manifolds; 2.2 Symplectomorphisms; 2.2.1 Physical transformations; 2.2.2 The switch map; 2.2.3 Hamiltonian transformations; 2.3 Examples of Hamiltonians; 2.3.1 The free particle and the geodesic flow; 2.3.2 Stereographic projection and the geodesic flow of the round metric; 2.3.3 Mechanical Hamiltonians

2.3.4 Magnetic Hamiltonians2.3.5 Physical symmetries; 2.3.6 Normal forms; 2.4 Hamiltonian structures; 2.5 Contact forms; 2.6 Liouville domains and contact type hypersurfaces; 2.7 Real Liouville domains and real contact manifolds; Chapter 3 Symmetries; 3.1 Poisson brackets and Noether's theorem; 3.2 Hamiltonian group actions and moment maps; 3.3 Angular momentum, the spatial Kepler problem, and the Runge-Lenz vector; 3.3.1 Central force: conservation of angular momentum; 3.3.2 The Kepler problem and its integrals; 3.3.3 The Runge-Lenz vector: another integral of the Kepler problem

3.4 Completely integrable systems3.5 The planar Kepler problem; Chapter 4 Regularization of Two-Body Collisions; 4.1 Moser regularization; 4.2 The Levi-Civita regularization; 4.3 Ligon-Schaaf regularization; 4.3.1 Proof of some of the properties of the Ligon-Schaaf map; Chapter 5 The Restricted Three-Body Problem; 5.1 The restricted three-body problem in an inertial frame; 5.2 Time-dependent transformations; 5.3 The circular restricted three-body problem in a rotating frame; 5.4 The five Lagrange points; 5.5 Hill's regions; 5.6 The rotating Kepler problem

5.7 Moser regularization of the restricted three-body problem5.8 Hill's lunar problem; 5.8.1 Derivation of Hill's lunar problem; 5.8.2 Hill's lunar Hamiltonian; 5.9 Euler's problem of two fixed centers; Chapter 6 Contact Geometry and the Restricted Three-Body Problem; 6.1 A contact structure for Hill's lunar problem; 6.2 Contact connected sum; 6.2.1 Contact version; 6.3 A real contact structure for the restricted three-body problem; Chapter 7 Periodic Orbits in Hamiltonian Systems; 7.1 A short history of the research on periodic orbits; 7.2 Variational approach; 7.3 The kernel of the Hessian

7.4 Periodic orbits of the first and second kind7.5 Symmetric periodic orbits and brake orbits; 7.6 Blue sky catastrophes; 7.7 Elliptic and hyperbolic orbits; Chapter 8 Periodic Orbits in the Restricted Three-Body Problem; 8.1 Some heroes in the search for periodic orbits; 8.2 Periodic orbits in the rotating Kepler problem; 8.2.1 The shape of the orbits if; 8.2.2 The circular orbits; 8.2.3 The averaging method; 8.2.4 Periodic orbits of the second kind; 8.3 The retrograde and direct periodic orbit; 8.3.1 Low energies; 8.3.2 Birkhoff's shooting method; 8.3.3 The Birkhoff set