R. Martinez
Barcelona, Spain
Dept. de Matematiques Universitat Autonoma de Barcelona, Bellaterra
Publications:
Martinez R., Simó C.
Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion
2014, vol. 19, no. 6, pp. 745765
Abstract
Invariant manifolds of a periodic orbit at infinity in the planar circular RTBP are studied. To this end we consider the intersection of the manifolds with the passage through the barycentric pericenter. The intersections of the stable and unstable manifolds have a common even part, which can be seen as a displaced version of the twobody problem, and an odd part which gives rise to a splitting. The theoretical formulas obtained for a Jacobi constant $C$ large enough are compared to direct numerical computations showing improved agreement when $C$ increases. A return map to the pericenter passage is derived, and using an approximation by standardlike maps, one can make a prediction of the location of the boundaries of bounded motion. This result is compared to numerical estimates, again improving for increasing $C$. Several anomalous phenomena are described.

Martinez R., Simó C.
NonIntegrability of Hamiltonian Systems Through High Order Variational Equations: Summary of Results and Examples
2009, vol. 14, no. 3, pp. 323348
Abstract
This paper deals with nonintegrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order $k$ variational equations, for arbitrary values of $k$, to prove nonintegrability. Moreover, using third order variational equations we prove the nonintegrability of a nonlinear springpendulum problem for the values of the parameter that can not be decided using first order variational equations.
