Hill problem analytical theory to the order four: application to the computation of frozen orbits around planetary satellites

Parte de este trabajo ha sido presentado en el 20th International Symposium on Space Flight Dynamics, Annapolis, Maryland, USA, 24–28 de septiembre de 2007.Frozen orbits of the Hill problem are determined in the double-averaged problem, where short and long-period terms are removed by means of Lie transforms. Due to the perturbation method we use, the initial conditions of corresponding quasi-periodic solutions in the nonaveraged problem are computed straightforwardly. Moreover, the method provides the explicit equations of the transformation that connects the averaged and nonaveraged models. A fourth-order analytical theory is necessary for the accurate computation of quasi-periodic frozen orbits.This work was supported from Projects ESP 2007-64068 the first author and MTM 2008-03818 the second author of the Ministry of Science and Innovation of Spain is acknowledged

On co-orbital quasi-periodic motion in the three-body problem

Within the framework of the planar three-body problem we establish the existence of quasi-periodic motions and KAM $4$-tori related to the co-orbital motion of two small moons about a large planet where the moons move in nearly circular orbits with almost equal radii. The approach is based on a combination of normal form and symplectic reduction theories and the application of a KAM theorem for high-order degenerate systems. To accomplish our results we need to expand the Hamiltonian of the three-body problem as a perturbation of two uncoupled Kepler problems. This approximation is valid in the region of phase space where co-orbital solutions occur.Peer ReviewedPostprint (author's final draft

Qualitative features of Hamiltonian systems through averaging and reduction

In this work we analyze the existence and stability of periodic solutions to a Hamiltonian vector field which is a small perturbation of a vector field tangent to the fibers of a circle bundle. By averaging the perturbation over the fibers of the circle bundle one obtains a Hamiltonian system on the reduced (orbit) space of the circle bundle. First we state results which have hypotheses on the reduced system and have conclusions about the full system. The second part is devoted to the application of the general results to the spatial lunar problem of celestial mechanics, i.e. the restricted three-body problem where the infinitesimal is close to one of the primaries. After scaling, the lunar problem is a perturbation of the Kepler problem, which after regularization is a circle bundle flow. We prove the existence of four families of periodic solutions for any small regular perturbation of the spatial Kepler problem: we find the classical near circular periodic solutions and the near rectilinear periodic solutions for all values of the small parameter. Finally we compute their approximate multipliers.Ministerio de Educación y CienciaDepartamento de Educación y Cultura (Gobierno de Navarra

Transition State Theory For Laser-Driven Reactions

Recent developments in transition state theory brought about by dynamical systems theory are extended to time-dependent systems such as laser-driven reactions. Using time-dependent normal form theory, the authors construct a reaction coordinate with regular dynamics inside the transition region. The conservation of the associated action enables one to extract time-dependent invariant manifolds that act as separatrices between reactive and nonreactive trajectories and thus make it possible to predict the ultimate fate of a trajectory. They illustrate the power of our approach on a driven Henon-Heiles system, which serves as a simple example of a reactive system with several open channels. The present generalization of transition state theory to driven systems will allow one to study processes such as the control of chemical reactions through laser pulses