This paper brings a unifying point of view for several problems of nonlinear particle dynamics (including the classical Kepler, Coulomb, and Manev problems) by using the qualitative theory of dynamical systems. We consider two-body problems with potentials of the type A=r + B=r 2 , where r is the distance between particles, and A; B are real constants. Using McGehee-type transformations and exploiting the rotational symmetry speciÿc to this class of vector ÿelds, we set the equations of motion in a reduced phase space and study all possible choices of the constants A and B. In this new setting the dynamics appears elegant and simple. In the end we use the phase-space structure to tackle the question of block-regularizing collisions, which asks whether orbits can be extended beyond the collision singularity in a physically meaningful way, i.e. by preserving the continuity of the general solution with respect to initial data. HistoryThe above class of problems is over three centuries old. Newton was the ÿrst to consider central-force and two-body problems in his monumental work Principia, whose

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.

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Several contemporary modified models of gravity predict the existence of a non-Newtonian Yukawa-type correction to the classical gravitational potential. We study the motion of a secondary celestial body under the influence of the corrected gravitational force of a primary. We derive two equations to approximate the periastron time rate of change and its total variation over one revolution (i.e., the difference between the anomalistic period and the Keplerian period) under the influence of the non-Newtonian radial acceleration. Kinematically, this influence produces apsidal motion. We performed numerical estimations for Mercury, for the companion star of the pulsar PSR 1913+16, and for the extrasolar Planet b of the star HD 80606. We also considered the case of the artificial Earth satellite GRACE-A, but the results present a low degree of reliability from a practical standpoint.