Regularization of restricted 3-body problems. (English) Zbl 1351.70012

Castilho, César; Vidal, Claudio

Qual. Theory Dyn. Syst. 7, No. 2, 451-466 (2009).

Summary: Using geometric mechanics we regularize the binary collisions of planar restricted three-body problems (with zero and nonzero angular momentum). The regularization is general enough to allow the primaries to move under any arbitrary dynamics and has as special cases all the classical restricted three-body problems, namely; circular, elliptic, parabolic and hyperbolic. This generalizes the classical procedures of T. Levi-Civita [Acta Math. 42, 99–144 (1920; JFM 47.0837.01)], G. D. Birkhoff [Ann. Sc. Norm. Super. Pisa, II. Ser. 4, 267–306 (1935; Zbl 0012.12804, JFM 61.1480.03)], T. N. Thiele [Astron. Nachr. 138, 1–10 (1895; JFM 26.1098.01)] and C. Burrau [Vierteljahrsschr. Astron. Ges. 41, 261–266 (1906; JFM 37.0968.01)]. Explicit formulas are given for the regularizations.

MSC:

70F07	Three-body problems
70F15	Celestial mechanics
70H15	Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics

Keywords:

restricted three-body problem; regularization; gravitational systems

Citations:

JFM 47.0837.01; Zbl 0012.12804; JFM 61.1480.03; JFM 26.1098.01; JFM 37.0968.01