Invariant manifolds of L_3 and horseshoe motion in the restricted three-body problem

In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (which surround three equilibrium points called L3, L4 and L5), when the mass parameter µ is positive and small; we describe the structure of such families from the two-body problem (µ = 0). On the other hand, the region of existence of horseshoe periodic orbits for any value of µ ∈ (0, 1/2] implies the understanding of the behaviour of the invariant manifolds of L3. So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L3, and on the simple infinite and double infinite period homoclinic phenomena are also analysed. Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe periodic orbits are considered in detail

On central configurations of the kn-body problem

We consider planar central configurations of the Newtonian kn-body problem consisting in k groups of regular n-gons of equal masses, called (k, n)-crown. We derive the equations of central configurations for a general (k, n)-crown. When k = 2 we prove the existence of a twisted (2, n)-crown for any value of the mass ratio. Moreover, for n = 3,4 and any value of the mass ratio, we give the exact number of twisted (2, n)-crowns, and describe their location. Finally, we conjecture that for any value of the mass ratio there exist exactly three (2, n)-crowns for n = 5.Peer ReviewedPostprint (author's final draft

Convex central configurations of two twisted n-gons

Extended Conference abstract corresponding to talk given at the "Conference on Hamiltonian Systems and Celestial Mechanics 2014" (HAMSYS2014) held at the Centre de Recerca Matemàtica (CRM) in Barcelona from June 2nd to 6th, 2014. The talk is about Central Configurations, Periodic Orbits and Hamiltonian Systems with applications to Celestial Mechanics - a very modern and active field of researc

On strictly convex central configurations of the 2n-body problem

We consider planar central configurations of the Newtonian 2n-body problem consisting in two twisted regular n-gons of equal masses. We prove the conjecture that for n=5 all convex central configurations of two twisted regular n-gons are strictly convex.Peer ReviewedPostprint (author's final draft

Pseudo-heteroclinic connections between bicircular restricted four-body problems

In this paper, we show a mechanism to explain transport from the outer to the inner Solar system. Such a mechanism is based on dynamical systems theory. More concretely, we consider a sequence of uncoupled bicircular restricted four-body problems –BR4BP –(involving the Sun, Jupiter, a planet and an infinitesimal mass), being the planet Neptune, Uranus and Saturn. For each BR4BP, we compute the dynamical substitutes of the collinear equilibrium points of the corresponding restricted three-body problem (Sun, planet and infinitesimal mass), which become periodic orbits. These periodic orbits are unstable, and the role that their invariant manifolds play in relation with transport from exterior planets to the inner ones is discussed.Peer ReviewedPostprint (published version

The dynamics around the collinear point L3 of the RTBP

We consider the Restricted Three Body Problem (RTBP), and we restrict our attention to the equilibrium point L3. Our aim is centered in the description, as global as possible, of the dynamics around this equilibrium point. In this communication, we initially consider small values of µ, for which homoclinic connections to the equilibrium point L3 are horseshoe-shaped, and then, other values of µ are considered. We compute the objects in the center manifold of L3, including the invariant manifolds associated with them. They are computed by purely numerical procedures, in order to avoid the convergence restrictions of the semi-analytical ones (typically used around L1 or L2). We deal with homoclinic connections of periodic orbits and develop some numerical tools in order to compute them. These tools can be extended to compute also homoclinic connections to invariant tori

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L3, as an example of a center×saddle equilibrium point in a Hamiltonian with two degrees of freedom.We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L3, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ1, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ1, corresponding to m-round SHO. Some comments on related analytical results are also made.Peer ReviewedPostprint (published version

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L3, as an example of a center×saddle equilibrium point in a Hamiltonian with two degrees of freedom.We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L3, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ1, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ1, corresponding to m-round SHO. Some comments on related analytical results are also made.Peer ReviewedPostprint (published version

Dinàmica de varietats espacials: les autopistes de l’univers

En aquest article volem il.lustrar com la comprensió de la dinàmica d’alguns models de la mecànica celeste permet explicar alguns fenòmens astronòmics i dissenyar missions realistes a l’espai. El model paradigmàtic usat és el problema restringit de tres cossos, en el qual els objectes que tenen un paper essencial són les varietats invariants de les anomenades òrbites de libració, és a dir, òrbites periòdiques i quasi periòdiques al voltant dels anomenats punts d’equilibri col.lineals del model.Descriurem alguns d’aquests fenòmens i esmentarem algunes missions concretes.Finalment, comentarem altres models també útils (i més sofisticats) a l’astrodinàmica i acabarem amb algun comentari de com les eines de sistemes dinàmics es poden traslladar del món macroscòpic (celeste) al microscòpic, com per exemple el de la física atòmica clàssica.Postprint (published version

Ejection–collision orbits in two degrees of freedom problems in celestial mechanics

The version of record is available online at: http://dx.doi.org/10.1007/s00332-021-09721-5In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of N-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection–collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2-dimensional invariant manifolds and on the behavior of the dynamics on the total collision manifold; both of them are thoroughly described.Peer ReviewedPostprint (author's final draft