0000000000139754
AUTHOR
J. Martínez Alfaro
Round-handle decomposition ofS2×S1
A round-handle decomposition is associated with a non-singular Morse–Smale flow on 3-manifolds prime to S 2× S 1. This decomposition has been built only for the 3-sphere S 3. In this paper we obtain the round-handle decomposition of non-singular Morse–Smale flows on S 2× S 1, in order to get all the different fattened round handles in this manifold. Some of them include non-separating boundary components that induce the topology of the links of periodic orbits.
Collision Orbits in the Isosceles Rectilinear Restricted Problem
In the study of the Collinear Three-Body Problem, McGehee (1974) introduced a new set of coordinates which had the effect of blowing up the triple collision singularity. Subsequently, his method has been used to analyze some other collision or singularities. Recently, Wang (1986) introduced another transformation which differs from the McGehee’s coordinates in the fact that the blowing-up factor is now the potential function, U, instead of the moment of inertia, I. Meyer and Wang (1993) have applied this method to the Restricted Isosceles Three-body Problem with positive energy and Cors and Llibre (1994) to the hyperbolic restricted three-body problem. In this paper we study the singulariti…
Links and Bifurcations in Nonsingular Morse–Smale Systems
Wada's theorem classifies the set of periodic orbits in NMS systems on S3 as links, that can be written in terms of six operations. This characterization allows us to study the topological restrictions that links require to suffer a given codimension one bifurcation. Moreover, these results are reproduced in the case of NMS systems with rotational symmetries, introducing new geometrical tools.
Bifurcations of Links of Periodic Orbits in Non-Singular Systems with Two Rotational Symmetries on S3
A topological characterization of all possible links composed of the periodic orbits of a Non Singular Morse-Smale flow on S3 has been made by M. Wada. The presence of symmetry forces the appearance of given types of links. In this paper we introduce a geometrical tool to represent these type of links when a symmetry around two axes is considered on NMS systems: mosaics. On the other hand, we use mosaics to study what kind of bifurcation can occur in this type of system maintaining the symmetry.
Orbital Structure of the Two Fixed Centres Problem
The set of orbits of the Two Fixed Centres problem has been known for a long time (Charlier, 1902, 1907; Pars, 1965), since it is an integrable Hamiltonian system.
Collision orbits in the oblate planet problem
Some of the properties of the oblate planet problem are derived. We use the technique of blowing up the singularity to study the collision orbits. We define some families of them in terms of their asymptotic behavior.
Bifurcations of links of periodic orbits in non-singular Morse–Smale systems with a rotational symmetry on S3
Abstract In this paper we consider a rotational symmetry on a non-singular Morse–Smale (NMS) system analyzing the restrictions this symmetry imposes on the links defined by the set of its periodic orbits and to the appearance of local generic codimension one bifurcations in the set of NMS flows on S 3 . The topological characterization is obtained by writing the involved links in terms of Wada operations. It is also obtained that symmetry implies that in general bifurcations have to be multiple. On the other hand, we also see that there exists a set of links that cannot be related to any other by sequences of this kind of bifurcation.
Bifurcations of Links of Periodic Orbits in Mathieu Systems
We prove that orbits escape from infinity, and that therefore the sphere S can be considered as its phase space. If the parameter δ is large enough, the system is non-singular MorseSmale, and its periodic orbits define a Hopf link. As δ decreases, the system undergoes some bifurcations that we describe geometrically. We relate the bifurcation orbits to periodic orbits continued from the linear Mathieu equation.
Ejection and collision orbits of the spatial restricted three-body problem
We begin by describing the global flow of the spatial two body rotating problem, μ=0. The remainder of the work is devoted to study the ejection and collision orbits when μ>-0. We make use of the ‘blow up’ techniques to show that for any fixed value of the Jacobian constant the set of these orbits is diffeomorphic to S2×R. Also we find some particular collision-ejection orbits.
Types of Motion in the Oblate Planet Problem
We consider a mass point in the gravitational field of an oblate planet and in a meridianal plane. The Hamiltonian of the problem is: $$ \frac{1}{2}\left( {p_r^2 + \frac{{p_{\theta }^2}}{{{r^2}}}} \right) - \frac{1}{r} - \frac{\varepsilon }{{{r^3}}}\left( {1 - 3{{\sin }^2}\theta } \right) $$ .
KNOTS AND LINKS IN INTEGRABLE HAMILTONIAN SYSTEMS
The main purpose of this paper is to prove that Bott integrable Hamiltonian flows and non-singular Morse-Smale flows are closely related. As a consequence, we obtain a classification of the knots and links formed by periodic orbits of Bott integrable Hamiltonians on the 3-sphere and on the solid torus. We also show that most of Fomenko's theory on the topology of the energy levels of Bott integrable Hamiltonians can be derived from Morgan's results on 3-manifolds that admit non-singular Morse-Smale flows.
Persistence of Asteroids After a Close Encounter
We present here a first approximation to the planar circular restricted 2 + 2 problem. In this four body problem, we consider that the two secondaries do not affect the primaries but they do influence each other. It can be seen as a model for near collision orbits of two asteroids if the primaries are the Sun and Jupiter ([3], [5]). In particular, we analyze the values of the Jacobi constant of the two asteroids before and after the close approach.
Classes of orbits in the main problem of satellite theory
We consider the main problem in satellite theory restricted to the polar plane. For suitable values of the energy the system has two unstable periodic orbits. We classify the trajectories in terms of their ultimate behavior with respect these periodic orbits in: oscillating, asymptotic and capture orbits. We study the energy level set and the existence and properties of the mentioned types of motion.
Bifurcations of links of periodic orbits in non-singular Morse - Smale systems on
The set of periodic orbits of a non-singular Morse - Smale (NMS) flow on defines a link; a characterization of all possible links of NMS flows on has been developed by Wada. In the frame of codimension-one bifurcations, this characterization allows us to study the restrictions a link requires for suffering a given bifurcation. We also derive the topological description of the new link and the possibility of relating links by a chain of this type of bifurcation.
Invariant rotational curves in Sitnikov's Problem
The Sitnikov's Problem is a Restricted Three-Body Problem of Celestial Mechanics depending on a parameter, the eccentricity,e. The Hamiltonian,H(z, v, t, e), does not depend ont ife=0 and we have an integrable system; ife is small the KAM Theory proves the existence of invariant rotational curves, IRC. For larger eccentricities, we show that there exist two complementary sequences of intervals of values ofe that accumulate to the maximum admissible value of the eccentricity, 1, and such that, for one of the sequences IRC around a fixed point persist. Moreover, they shrink to the planez=0 ase tends to 1.