Global study of the 2d secular 3-body problem

Following the same central idea of Féjoz [9] [10] [8], we study the planar averaged $3$-body problem without making use of series developments, as is usual, but instead we perform a global geometric analysis: the space of the orbits for a fixed energy is reduced under the rotational symmetry to a $2$-dimensional symplectic manifold, where the motion is described by the level curves of the reduced Hamiltonian. The number and location of the critical points are investigated both analytically and numerically, confirming a conjecture of Féjoz.