Ernesto A. Lacomba

Gradient vector fields on cosymplectic manifolds

The authors study some geometrical properties of gradient vector fields on cosymplectic manifolds, thereby emphasizing the close analogy with Hamiltonian systems on symplectic manifolds. It is shown that gradient vector fields and, more generally, local gradient vector fields can be characterized in terms of Lagrangian submanifolds of the tangent bundle with respect to an induced symplectic structure. In addition, the symmetry and reduction properties of gradient vector fields are investigated.

On the anisotropic Manev problem

We consider the Manev potential, given by the sum between the inverse and the inverse square of the distance, in an anisotropic space, i.e., such that the force acts differently in each direction. Using McGehee coordinates, we blow up the collision singularity, paste a collision manifold to the phase space, study the flow on and near the collision manifold, and find a positive-measure set of collision orbits. Besides frontal homothetic, frontal nonhomothetic, and spiraling collisions and ejections, we put into the evidence the surprising class of oscillatory collision and ejection orbits. Using the infinity manifold, we further tackle capture and escape solutions in the zero-energy case. By finding the connection orbits between equilibria and/or cycles at impact and at infinity, we describe a large class of capture-collision and ejection-escape solutions.

The global flow of the Manev problem

The Manev problem (a two‐body problem given by a potential of the form A/r+B/r2, where r is the distance between particles and A,B are positive constants) comprises several important physical models, having its roots in research done by Isaac Newton. We provide its analytic solution, then completely describe its global flow using McGehee coordinates and topological methods, and offer the physical interpretation of all solutions. We prove that if the energy constant is negative, the orbits are, generically, precessional ellipses, except for a zero‐measure set of initial data, for which they are ellipses. For zero energy, the orbits are precessional parabolas, and for positive energy they are precessional hyperbolas. In all these cases, the set of initial data leading to collisions has positive measure.

Lagrangian submanifolds and higher-order mechanical systems

Higher-order Lagrangian and Hamiltonian systems (time dependent or independent) are interpreted in terms of Lagrangian submanifolds of sympletic higher-order tangent bundles. The relation between both formalisms is given.

Boundary manifolds for energy surfaces in celestial mechanics

We complete Mc Gehees picture of introducing a boundary (total collision) manifold to each energy surface. This is done by constructing the missing components of its boundary as other submanifolds. representing now the asymptotic behavior at infinity.It is necessary to treat each caseh=0,h>0 orh<0 separately. In the first case, we repeat the known result that the behavior at total escape is the same as in total collision. In particular, we explain why the situation is radically different in theh>0 case compared with the zero energy case. In the caseh<0 we have many infinity manifold components. and the general situation is not quite well understood.Finally, our results forh≥0 are shown to be valid for general homogeneous potentials.

Geometric formulation of Carnot's theorem

A geometrical approach for one-sided constraints is given. The Riemannian metric is used in order to define convenient projectors which give the post-impulses in terms of the pre-impulses. A formulation of Carnots theorem within this geometric framework is exhibited.

Analysis of some degenerate quadruple collisions

We consider the trapezoidal problem of four bodies. This is a special problem where only three degrees of freedom are involved. The blow up method of McGehee can be used to deal with the quadruple collision. Two degenerate cases are studied in this paper: the rectangular and the collinear problems. They have only two degrees of freedom and the analysis of total collapse can be done in a way similar to the one used for the collinear and isosceles problems of three bodies. We fully analyze the flow on the total collision manifold, reducing the problem of finding heteroclinic connections to the study of a single ordinary differential equation. For the collinear case, from which arises a one parameter family of equations, the analysis for extreme values of the parameter is done and numerical computations fill up the gap for the intermediate values. Dynamical consequences for possible motions near total collision as well as for regularization are obtained.

Regularization of simultaneous binary collisions in the n-body problem

Abstract In the n -body problem a collision singularity occurs when the positions of two or more bodies tend to coincide. It is well known that binary collisions can be regularized and collisions of three or more bodies cannot be regularized, unless we restrict the study to some simple class of problem. In this paper we show that the remaining case, that is, several binary collisions occurring simultaneously, is also regularizable.

New Advances in Celestial Mechanics and Hamiltonian Systems

Exchange and capture in the planar restricted parabolic 3-body problem.- Fitting Invariant Curves on Billiard Tables and the Birkhoff-Herman Theorem.- Construction of Periodic Orbits in Hills Problem for C ? 34/3.- Are there perverse choreographies?.- Blow up of total collision in the tetrahedral non-rotating four body problem.- Regularization of single binary collisions.- A Survey on Bifurcations of Invariant Tori.- Perturbing the Lagrange solution to the general three body problem.- Horseshoe periodic orbits in the restricted three body problem.- Instability of Periodic Orbits in the Restricted Three Body Problem.- Syzygies and the Integral Manifolds of the Spatial N-Body Problem.- Dynamics and bifurcation near the transition from stability to complex instability.- Invariant Manifolds of Spatial Restricted Three-Body Problems: the Lunar Case.- Path Integral Quantization of the Sphere.- Non-holonomic systems with symmetry allowing a conformally symplectic reduction.- 253.

Origin and Infinity Manifolds for Mechanical Systems with Homogeneous Potentials

We study manifolds describing the behavior of motions close to the origin and at infinity of configuration space, for mechanical systems with homogeneous potentials. We find an inversion between these behaviors when the sign of the degree of homogeneity is changed. In some cases, the blow up equations can be written in canonical form, by first reducing to a contact structure. A motivation for the use of blow-up techniques is given, and some examples are studied in detail.