Vasile Mioc

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.

Phase-space structure and regularization of Manev-type problems

Phase-space structure and regularization of Manev-type problems Florin Diacu a;∗, Vasile Mioc b, Cristina Stoica c a Department of Mathematics and Statistics, University of Victoria, Victoria, Canada V8W 3P4 b Astronomical Institute of the Romanian Academy, Astronomical Observatory Cluj-Napoca, Str. Cire silor 19, 3400 Cluj-Napoca, Romania c Institute for Gravitation and Space Sciences, Laboratory for Gravitation, Str. Mendeleev 21-25, Bucharest, Romania

The Schwarzschild Problem in Astrophysics

The Schwarzschild problem (the two-body problem associated to apotential of the form A/r + B/r3 has been qualitativelyinvestigated in an astrophysical framework, exemplified by two likelysituations: motion of a particle in the photogravitational field ofan oblate, rotating star, or in that of a star which generates aSchwarzschild field. Using McGehee-type transformations, regularizedequations of motion are obtained, and the collision singularity isblown up and replaced by the collision manifold λ (a torus)pasted on the phase space. The flow on λ is fullycharacterized. Then, reducing the 4D phase space to dimension 2, theglobal flow in the phase plane is depicted for all possible values ofthe energy and for all combinations of nonzero A and B. Eachphase trajectory is interpreted in terms of physical motion,obtaining in this way a telling geometric and physical picture of themodel.

On the Schwarzschild-Type Polygonal ( n + 1 ) - Body Problem and on the Associated Restricted Problem

The planar symmetrical (n-fl)-body problem in a Schwarzschild-type field is being investigated. One proves that, if η equal masses are initially situated at the vertices of a regular polygon centered in the (n+l)th mass, and if the initial velocities form a vector field symmetrical with respect to the central mass, then the polygonal configuration is preserved all along the motion, but with variable side and with variable rotation around the center. The motion of every mass relative to the center is given by the solution of the Schwarzschild-type two-body problem. All possible behaviors of the polygonal solution are surveyed. In the second part of the paper, the relative equilibria of the (n+l)-body problem are pointed out. One associates a restricted problem to them, for which the Jacobi integral is proved to exist.

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.

Periodic motions around pulsating stars

The periodic motion of a test particle (dust, grain, or a larger body) around a pulsating star with a luminosity oscillation of small amplitude (featured by a small parameterB) is being studied. The perturbations of all orbital elements are determined to first order inB, by using Delaunay-type canonical variables and a method whose bases were put forth by von Zeipel. According to the value of the ratio oscillation frequency/dynamic frequency, three possible situations are pointed out: nonresonant (NR), quasi-resonant (QR), and resonant (R). The solution of motion equations shows that only in the (QR) and (R) cases there are orbital parameters (argument of periastron and mean anomaly) affected by secular perturbations. These solutions (which indicate a secularly stable motion in a first approximation) are valid over prediction times of orderB−1 in the (NR) case andB−1/2 in the (QR) and (R) cases. The theory may be applied to various astronomical situations.

Exact solutions for a class of integrable Hénon-Heiles-type systems

We study the exact solutions of a class of integrable Henon–Heiles-type systems (according to the analysis of Bountis et al. [Phys. Rev. A 25, 1257 (1982)]). These solutions are expressed in terms of two-dimensional Kleinian functions. Special periodic solutions are expressed in terms of the well-known Weierstrass function. We extend some of our results to a generalized Henon–Heiles-type system with (n+1) degrees of freedom.

Binary collisions in quasihomogeneous fields

Abstract The quasihomogeneous fields (featured by sums of homogeneous potentials) model a lot of concrete fields met in problems of nonlinear particle dynamics. The particularly important case of binary collisions in such fields is tackled, and the local behaviour of the corresponding solutions is fully understood.

The periodic Gyldén‐type problem in astrophysics

We present some analytical and numerical results for the resonant Gylden‐type problem and identify some astrophysical situations for which this model can be useful. For small‐amplitude periodic variation, we study the behavior of the system in the neighborhood of resonances. Considering the simplest case of the variation, we perform some numerical experiments. The phase portraits are very complex: a mixture of oscillation zones, circulation zones and chaotic zones.

Symmetries in the two-body problem associated to quasihomogeneous potentials

Abstract The dynamics in quasihomogeneous fields (characterized by sums of homogeneous potentials) models a lot of concrete physical and astronomical situations. The two-body problem in Cartesian and standard polar coordinates, as well as in regularizing McGehee-type coordinates of both collision-blow-up kind and infinity-blow-up kind, is being tackled. The corresponding vector fields present symmetries that form isomorphic Abelian groups endowed with an idempotent structure.