Infinity Manifold Research Articles

Qualitative analysis of the anisotropic two-body problem under Seeliger's potential

In this paper we study the two-body problem that describes the motion of two-point masses in an anisotropic space under the influence of the Seeliger potential. We will show that the set of initial conditions leading to collisions and ejections has positive measure and study the capture and escape solutions in the positive-energy case using the infinity manifold. We will also apply the Melnikov method to show that the flow on the zero-energy manifold of another potential which is the sum of the classical Keplerian potential and the anisotropic Seeliger's potential perturbation is chaotic.

Qualitative analysis of the anisotropic two-body problem with relativistic potential

In this paper we study the two-body problem that describes the motion of two-point masses in an anisotropic space under the influence of a Newtonian force-law with two relativistic correction terms. We will show that the set of initial conditions leading to collisions and ejections have positive measure and study the capture and escape solutions in the zero-energy case using the infinity manifold. We will also apply the Melnikov method to show that the flow on the zero-energy manifold of another potential which is the sum of the classical Keplerian potential and two anisotropic perturbation which also take into account two relativistic correction terms is chaotic.

Qualitative analysis of the anisotropic two-body problem with generalized Lennard-Jones potential

This paper continue the study of the generalized Lennard-Jones potential started in Bărbosu et al. (J Math Chem 49(9):1961–1975, 2011) for a more general situation. More precisely we study the two-body problem with generalized Lennard-Jones potential in an anisotropic space. We will show that the set of initial conditions leading to collisions and ejections have positive measure. We also study the capture and escape solutions in the zero-energy case using the infinity manifold. We also show that the flow on the zero energy manifold of a two-body problem given by the sum of the Newtonian potential and the two anisotropic perturbations corresponding to the generalized Lennard-Jones potential is chaotic.

ON THE ANISOTROPIC POTENTIALS OF MANEV–SCHWARZSCHILD TYPE

In this paper, we study the two-body problem that describes the motion of two-point masses in an anisotropic space under the influence of a Newtonian force-law with two relativistic correction terms. We will show that the set of initial conditions leading to collisions and ejections and leading to escapes and captures have positive measure. Using the infinity manifold, we study capture and escape solutions in the zero-energy case. We also show that the flow on the zero energy manifold of a two-body problem given by the sum of the Newtonian potential and three anisotropic perturbations (corresponding to three relativistic correction terms) is chaotic.

The Flow on Negative Energy of Mechanical Systems with Homogeneous Polynomial Potentials

In this paper we study the global flow of classical mechanical systems with homogeneous polynomial potentials with two degrees of freedom by using McGehee coordinates. Then the flow is extended to a 3-dimensional manifold whose boundary is known as the infinity manifold. In the negative energy case, this manifold is compact, it can be imbedded in \({{{\mathbb R}^3}}\) , and its shape is generically described for any degree of homogeneity. We study the global flow for the non-trivial case corresponding to a “peapod” shaped manifold which appears for homogeneity degree greater than four.

Operad profiles of Nijenhuis structures

Recently S. Merkulov [S.A. Merkulov, Operads, deformation theory and F-manifolds, in: Frobenius manifolds, in: Aspects Math., vol. E36, Vieweg, Wiesbaden, 2004, pp. 213–251; S.A. Merkulov, Nijenhuis infinity and contractible differential graded manifolds, Compos. Math. 141 (5) (2005) 1238–1254; S.A. Merkulov, Prop profile of Poisson geometry, Comm. Math. Phys. 262 (2006) 117–135] established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on the Koszul dual of a certain quadratic operad. In this paper we prove, using the PBW-basis method of E. Hoffbeck [E. Hoffbeck, A Poincaré–Birkhoff–Witt criterion for Koszul operads, arXiv:0709.2286v3 [math.AT], 2008], that the operad governing Nijenhuis structures is Koszul, thereby showing that Nijenhuis structures correspond to representations of the minimal resolution of this operad. We also construct an operad such that representations of its minimal resolution in a vector space V are in one-to-one correspondence with pairs of compatible Nijenhuis structures on the formal manifold associated to V.

Collision and escape orbits in a generalized Hénon–Heiles model

The motion of a material point of unit mass in a field determined by a generalized Hénon–Heiles potential U=Aq12+Bq22+Cq12q2+Dq23, with (q1,q2)= standard Cartesian coordinates and (A,B,C,D)∈(0,∞)2×R2, is addressed for two limit situations: collision and escape. Using McGehee-type transformations, the corresponding collision and infinity boundary manifolds pasted on the phase space are determined. These are fictitious manifolds, but, due to the continuity with respect to initial data, their flow determines the near by orbit behaviour.The dynamics on the collision and infinity manifolds is fully described. The topology of the flow on the collision manifold is independent of the parameters. In the full phase space, while spiraling collision orbits are present, most of the orbits avoid collision. The topology of the flow on the infinity manifold changes as the ratio between C and D varies. More precisely, there are two symmetric pitchfork bifurcations along the line 2C−3D=0, due to the reshaping of the potential along the bifurcation line. Besides rectilinear and spiraling orbits, the near-escape dynamics includes oscillatory orbits, for which angular momentum alternates sign.

Nijenhuis infinity and contractible differential graded manifolds

we find a minimal differential graded (dg) operad whose generic representations in which satisfy the nijenhuis integrability condition. this operad is of a surprisingly simple origin: it is the cobar construction on the quadratic operad of homologically trivial dg lie algebras. as a by-product we obtain a strong-homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifolds.

The Kepler problem with anisotropic perturbations

We study a two-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree −β, β⩾2. For β>2, the sets of initial conditions leading to collisions∕ejections and the one leading to escapes∕captures have positive measure. For β>2 and β≠3, the flow on the zero-energy manifold is chaotic. For β=2, a case we prove integrable, the infinity manifold of the zero-energy level has heteroclinic connections with the collision manifold.

The Anisotropic Schwarzschild-Type Problem, Main Features

The two-body problem associated to an anisotropic Schwarzschild-type field is being tackled. Both the motion equations and the energy integral are regularized via McGehee-type transformations. The regular vector field exhibits nice symmetries that form a commutative group endowed with an idempotent structure. The physically fictitious flows on the collision and infinity manifolds, as well as the local flows in the neighbourhood of these manifolds, are fully described. Homothetic, spiral, and oscillatory orbits are pointed out. Some features of the global flow are depicted for all possible levels of energy. For the negative-energy case, few things have been done. The positive-energy global flow does not have zero-velocity curves; every orbit is of the type ejection – escape or capture – collision. In the zero-energy case, the collision and infinity manifolds have a very similar structure. The existence of eight trajectories that connect the equilibria on these manifolds is proved. The projectability of the zero-energy global flow completes the full understanding of the problem in this case.

Global dynamics of mechanical systems with cubic potentials

We study the behavior of solutions of mechanical systems with polynomial potentials of degree 3 by using a blow up of McGehee type. We first state some general properties for positive degree homogeneous potentials. In particular, we prove a very general property of transversality of the invariant manifolds of the flow along the homothetic orbit. The paper focuses in the study of global flow in the case of homogeneous polynomial potentials of degree 3 for negative energy. The flow is fairly simple because of its gradient-like structure, although for some values of the polynomial coefficients we have diverse behaviour of the separatrices on the infinity manifold, which are essential to describe the global flow.

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 zonal satellite problem - II: Near-escape flow

The study of the zonal satellite problem is continued by tackling the situation r??. New equations of motion (for which the infinite distance is a singularity) and the corresponding first integrals of energy and angular momentum are set up. The infinity singularity is blown up via McGehee-type transformations, and the infinity manifold is pasted on the phase space. The fictitious flow on this manifold is described. Then, resorting to the rotational symmetry of the problem and to the angular momentum integral, the near-escape local flow is depicted. The corresponding phase curves are interpreted as physical motions.

Infinity manifold of a swinging Atwood's machine

The unbounded motions of a swinging Atwood's machine are analysed by blowing up the singularity at infinity. The asymptotic motion is reduced to a gradient flow on an ellipsoid. By studying the flow on this ellipsoid it is shown that the unbounded orbits oscillate either an infinite number of times or not at all, depending only on a critical value of the mass ratio.

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.