Luis Franco-Pérez

We study a particular planar four body problem with three degrees of freedom, where the particles move under the influence of a potential U such that for = 1 we recover the classical Newtonian Potential. We describe the topology of the manifolds of constant energy, for any value of the total energy h. We also describe the total collision manifold and its topology.

Keywords: The restricted problem Exponential dichotomy Binary collisions regularized. 80: geometric theory of differential equations a b s t r a c t We study the restricted 3-body problem with the constriction of motion to the unit circle. First, we study the 2-body problem on the unit circle and give the explicit solutions for a regularized version of the equations of motion for any initial data. We classify the motions in elliptic, parabolic, hyperbolic type and an equilibrium state. Then, we analyze the restricted 3-body problem on the unit circle when the primary bodies are performing elliptic and hyperbolic motions. We show the existence of just one equilibrium state when the masses of primary bodies are equal and we exhibit the hyperbolic structure of this equilibrium point via an exponential dichotomy. In the last part we regularize the equations of motion. We show the global dynamics and some periodic solutions with its respective period.

We develop a regularization for binary collisions in some restricted 3-body problems moving in planar one-dimensional spaces with constant curvature. The main characteristic of the regularization is that it preserves the Hamiltonian structure of the equations and it regularizes all the binary collisions with just one transformation. We apply this global symplectic regularization to the 2-body problem on the unit circle and we show the global dynamics. Also, we tackle the restricted 3-body problem with one fixed center in the unit circle and we give the global dynamics for the case when it has two fixed centers.