A New Analysis of the Three-Body Problem

Numerical studies to detect chaotic motion in the full planar averaged three-body problem

“…We study, initially, the stable and unstable 5 manifolds of the hyperbolic fixed point (g h , G h ) of the Poincaré map P H , e and we claim (see Numerical Evidence 5.1 in [9]) that a Transverse 5 We recall that local stable and unstable manifolds associated to the hyperbolic fixed point x * of a map P are defined as Homoclinic Intersection between the stable and unstable manifolds for P H , e exists. 6 In fact, the two manifolds have a transverse intersection at the point (g h , G h ). Then, we consider different planes i h orthogonal to h at different points (g i , G i , r i ) on the curve.…”

“…In particular, from an analytical point of view it has been studied by using normal form theory in order to prove stability estimates. For a complete treatment of the subject we refer the reader to the articles [4,6,16,17]. At the same time, the problem has been studied from a numerical point of view in order to prove existence of chaotic motions under particular initial configurations.…”

Chaotic coexistence of librational and rotational dynamics in the averaged planar three--body problem

“…We specify that we deal with the Hamiltonian of the full three-body problem, where "full" is used as opposed to the so-called "restricted" problem; thus, the three bodies have no negligible masses and the motion of each one is ruled by mutual Newtonian gravitational attraction. The idea, already used in previous articles such as [3,4,6,7] and coming from papers [19][20][21] is to consider, in principle, no Newtonian interaction between the three bodies as dominant. We use new coordinates (see [19]) associated to the three bodies and through suitable rescalings of variables and time we write the new Hamiltonian describing the model as the sum of a Keplerian part ruling the motion of two of three bodies plus a part depending on all the three bodies.…”