The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities, escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches or a definite time.In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics, we study the fine structure of the region of initial conditions, in particular the chaotic scattering region.Keywords Three-body problem · Rectilinear three-body problem · Triple approaches · Schubart periodic orbit · Escapes · Ejections A. I. Martynova State Forest Technical Academy, Institutskij per. 5, St. Petersburg 194021, Russia 123 94 V. V. Orlov et al.
The dynamical evolution of 15 000 equal-mass triple systems with zero initial velocities (the free-fall three-body problem) is considered. The equations of motion are numerically integrated using regularization of binary and triple encounters. We find 170 triple systems which reach a state where the motions take place within a limited region of phase space during a long time. These regions are concentrated in the zones of regular motions in the vicinities of stable periodic orbits: the von Schubart orbit in the rectilinear problem, the Broucke orbit in the isosceles problem, and the 'Eight' orbit. The classification of such metastable orbits is suggested. A change of the types is found during the dynamical evolution of some metastable systems. The triple system leaves the metastable regime after some time, and its evolution is finished by the escape of one body.Key words: instabilities -celestial mechanics. I N T RO D U C T I O NIn different fields of science, evolutionary processes can be described with a good accuracy using ordinary differential equations. Often the problem can be reduced to the study of the behaviour of a dynamical system with a small number of degrees of freedom. The theory of such systems is given in many books (Whittaker 1904;Birkhoff 1927;Pars 1964;Arnold 1979;Lichtenberg & Lieberman 1983;Schuster 1984;Antonov 1985; Zaslavskij & Sagdeev 1988;Boccaletti & Pucacco 1996, 1997.Only in some special cases one can solve the equations of motion for such dynamical systems analytically. As a rule, dynamical systems are non-integrable. However, using numerical and analytical considerations of specific dynamical systems one can reveal various types of trajectories: stable and unstable periodic orbits, regular orbits, escape orbits, stochastic and semi-stochastic trajectories. Regular orbits with finite motion occur in the vicinities of stable periodic orbits. Near unstable periodic orbits the motions possess a complex tangled character, and so-called deterministic chaos is generated (Schuster 1984). Dvorak et al. (1998) discovered a new property of some trajectories -'stickiness' for regular motion regions around stable periodic orbits. In particular, such phenomena were found in the Sitnikov problem (Sitnikov 1960). In this restricted three-body problem the zero-mass body moves in the gravitational field of a binary system along the straight line perpendicular to the binary orbital plane and crosses the binary centre of mass. It is interesting to search for similar phenomena in the general three-body problem where all three masses are non-zero. E-mail: vor@astro.spbu.ru Three simple periodic orbits are known in triple systems (von Schubart 1956; Broucke 1979; Moore 1993;Chenciner & Montgomery 2000). These orbits are shown in Figs 1-3 in the barycentric reference frame. All three orbits are realized in equalmass non-rotating triple systems. The orbits are stable (Hénon 1977;Orlov, Petrova & Martynova 2002; Simó 2002). Then according to KAM theory, regular trajectories with finite motions must...
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