The global flow of the Manev problem

Abstract

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

“…For A¿0 the energy levels in the reduced phase space are portions of ellipsoids (h¡0), paraboloids (h = 0), and hyperboloids of one or two sheets or cones (h¿0), as we have already seen in [6]. The physical motions correspond to both radial and spiral ejection-collision, ejectionescape, capture-collision, capture-escape, periodic, or quasiperiodic orbits.…”

“…Papers like [3,5,16,24,25] applied modern mathematical results (like KAM theory, the Melnikov method, etc.) as well as classical techniques, or went into the physical and astronomical signiÿcance of the model; for details regarding the speciÿc results of these works see [6]. From the mathematical point of view, Manev's potential opens a new ÿeld of research; it has o ered up to now surprising results concerning the dynamics of gravitational particles, which disagree with the classical ones when the motion takes place in the neighborhood of singularities (see [5,6,8,9]).…”

“…The false impression that Newton had exhausted the problem, and that this model had nothing but historical value, persisted (especially among physicists) until very recently (see, for example, Moulton [6]). Their paper appeared as a natural consequence of the recent interest in Manev's potential.…”

Periodic orbits of the spatial anisotropic Manev problem

“…One of the advantages of the Manev problem over the Keplerian is that it explains the perihelion advance of the inner planets with the same accuracy as relativity, see [5], [9], [11], [12]- [15] and [17]- [19].…”

Singularuties of the Newtonian N-body problem

“…Но прямолинейное движение является предельным случаем сильно вытянутого эллип-са, поэтому непрерывность по начальным условиям выполняется. Однако в задаче двух тел Ма-нева с релятивистской поправкой к ньютонову потенциалу двойные столкновения могут проис-ходить на спиральных траекториях, которые не аппроксимируют эллипс [4], поэтому движение не является регуляризуемым по начальным данным. Трактовка регуляризации по Леви-Чивита имеет б´ольшее практическое значение, поскольку она позволяет изучать траектории, проходя-щие вблизи столкновений.…”