The Manev Two-Body Problem: Quantitative and Qualitative Theory

Diacu, Florin; Mioc, Vasile; Stoica, Cristina

doi:10.1142/9789812796417_0014

Cited by 21 publications

(30 citation statements)

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“…The collision singularity, which occurs if and only if r = 0, will be blown up by using McGehee-type transformations (for more details see [6,11]). By formally multiplying the energy integral (2:9) by r 2 (a rigorous justiÿcation of this step is given in [11]), we obtain the relation (rṙ) 2 + (r 2u ) 2 = hr 2 + 2Ar + 2B:…”

Section: Blow-up and Reductionmentioning

confidence: 99%

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Phase-space structure and regularization of~Manev-type problems

Diacu

Mioc

Stoica

2000

Nonlinear Analysis: Theory, Methods & Applications

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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

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Section: Blow-up and Reductionmentioning

confidence: 99%

“…We can, in fact, determine the ow near the collision manifold, as it was done in [11] for A; B¿0. However, that approach is restricted to a local description.…”

Section: Blow-up and Reductionmentioning

confidence: 99%

Phase-space structure and regularization of~Manev-type problems

Diacu

Mioc

Stoica

2000

Nonlinear Analysis: Theory, Methods & Applications

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“…As to Fig. 8f-h, they give rise to the phase curves (5,10,9), (5,SE,9), (5,9), respectively. It is easy to see that Figs.…”

Section: Sub-critical-energy Levelsmentioning

confidence: 52%

“…7 The global flow for critical-negative-energy level and nonzero angular momentum Figure 6c corresponds to the separatrix formed by the heteroclinic curves 3, 3 , 4, 4 , the trajectories 5 and 5 , and the two saddles UE 1 and UE 2 . As to their physical significance, this one is the same for the pairs (3, 7), (3 , 7 ), (5,4), (5 , 4 ) belonging to Figs. 7 and 5), respectively.…”

Section: Critical-energy Levelmentioning

confidence: 73%

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The Two-body Global Flow for Negative Energy Under the Schwarzschild–de Sitter Potential

Mioc

Pérez‐Chavela²

2009

Qual. Theory Dyn. Syst.

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Dynamics in the Schwarzschild Isosceles Three Body Problem

2014

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We emphasize the distinct features of the Schwarzschild model when compared to its Newtonian counterpart. We prove that, in contrast to the Newtonian case, on any level of energy the measure of the set on initial conditions leading to triple collision is positive. Further, whereas in the Newtonian problem triple collision is asymptotically reached only for zero angular momentum, in the Schwarzschild problem the triple collision is possible for non-zero total angular momenta (e.g., when two of the mass points spin infinitely many times around the centre of mass). This phenomenon is known in celestial mechanics as the black-hole effect and it is understood as an analogue in the classical context of the behaviour near a Schwarzschild black hole. Also, while in the Newtonian problem all triple collision orbits are necessarily homothetic, in the Schwarzschild problem this is not necessarily true. In fact, in the Schwarzschild problem there exist triple collision orbits which are neither homothetic, nor homographic.

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The Manev Two-Body Problem: Quantitative and Qualitative Theory

Cited by 21 publications

References 0 publications

Phase-space structure and regularization of~Manev-type problems

Phase-space structure and regularization of~Manev-type problems

The Two-body Global Flow for Negative Energy Under the Schwarzschild–de Sitter Potential

Dynamics in the Schwarzschild Isosceles Three Body Problem

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