Three-body problem, restricted

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

F. Gabern and A. Jorba, Generalizing the restricted three-body problem the bianular and tricircular coherent problems, Astron. Astrophys. 420, 751 (2004). 

The restricted three-body problem Two bodies of finite masses, called primaries, revolve around their common center of mass in circular orbits and a third body with negligible mass moves under their gravitational attraction, but does not affect the orbits of the two primaries. In most astronomical applications the second primary has a small mass compared to the first primary, and consequently the motion of the third, massless, body is a perturbed Keplerian orbit. This is a model for the study of an asteroid (Jupiter being the second primary), a trans-Neptunian object (Neptune being the second primary) or an Earth-like planet in an extrasolar planetary system. 

Abstract Three-body systems with Hill-type stability are the generalization to the general three-body problem of the Hill-stable orbits of the circular restricted three-body problem. 

These systems have always a negative energy integral h and a large angular momentum c (in the axes of the center of masses), they are characterized by a product he2 smaller than or equal to that of the corresponding circular Euler motion with the same three masses. They have a close binary and a third body that can neither approach nor disrupt the close binary (well defined limit distances can be given in terms of the three masses and the initial conditions). However, and this is a major difference with the circular restricted three-body problem, the third body can sometimes escape to infinity. 

In a two body problem a and p are the usual semi-major axis and semi-latus rectum of the relative Keplerian orbit, and they are also these same elements of the relative orbit of the primaries in a restricted three-body problem. 

The classical Hill s curves of the restricted three-body problem are given in Figure 3 for equal primaries. The curves of constant ratio g/v are given in Figures 4 and 5 in terms of the position of mi with respect to m2 and m3 for the following mass ratios 2mi= m2 = m3 (Figure 4) and mi = m2 = m3 (Figure 5). 

L5 the three-body system of interest cannot approach equilateral configurations. Furthermore if 1Jp/a is larger than the value of q/v at the saddle points the zone of possible motion becomes disconnected into two or three parts and we reach then the extension of Hill stability of the circular restricted three-body problem to the general three-body problem. 

Sandor, Z., Erdi, B. and Efthymiopoulos, C. (2000). The Phase Space Structure Around L4 in the Restricted Three-Body Problem. Celestial Mechanics and Dynamical Astronomy, 78 113-123. 

Let S be a body with an infinitesimal mass, subject to the gravitational attraction of Pi, Psame plane (i.e., we neglect the relative inclinations) and that the trajectories of P1 and P2 are circular with... 

The regularization of the planar, circular, restricted three-body problem... 

In the framework of the circular, restricted three-body problem, let us consider the motion in the 3-dimensional space of the three bodies S, Pi and P2. The primaries move in the c/ic/2 plane around their common center of mass, while in the synodic frame their coordinates become Pi(p2,0,0), P2(—Pu 0) 0)- Assume that the (/lfjg-plane rotates with unit angular velocity about the vertical axis. Then, the Hamiltonian function is given by... 

Let us consider two bodies Pi, P2 with masses 1 — fi, fi, moving on circular orbits around the barycenter O. Let us normalize to unity their distance. In the framework of the circular, restricted three-body problem, let us consider the motion of a third body S, moving in the same plane of the primaries. Let its coordinates be ( 1, 2) in the synodic reference frame. 

Consider a rotating physical plane parametrized by the complex variable y e C for convenience we assume the fixed primaries of the restricted three-body problem to be situated at the points A, C given by the complex posititons y = — 1 and y = 1, respectively (see Figure 5). The complex variable of the parametric plane will be denoted by v and will be normalized in such a way that the primaries are mapped to v = -1 or v = 1, respectively. 

The contents of this paper include, with variable emphasis, the topics of a series of lectures whose main title was Routes to Order Capture into resonance . This was indeed the subject of the last section above. The study of this subject has, however, shown that - unlike the restricted three-body problem - capture into resonance drives the system immediately to stationary solutions known as Apsidal corotations . The whole theory of these solutions was also included in the paper from the beginning - that is, from the formulation of the Hamiltonian equations of the planetary motions and the expansion of the disturbing function in the high-eccentricity planetary three-body problem. The secular theory of non-resonant systems was also given. Motions with aligned or anti-aligned periapses, resonant or not, resulting from non-conservative processes (tidal interactions with the disc) in the early phases of the life of the system, seem to be frequent in extra-solar planetary systems. 

The most celebrated problem in celestial mechanics is the so-called three-body problem. First elucidated by Lagrange, this problem focuses on the determination of the allowed class of periodic motions for a massless particle orbiting a binary system. In this case, the motion is determined by the gravitational and centrifugal accelerations and also the Coriolis force. A closed form analytic solution is possible in only one case, that of equal masses in a circular orbit. This so-caUed restricted three-body problem can be specified by the curves of constant potential, also called the zero velocity surfaces. Consider a binary with a coplanar orbit for the third mass. In this case, a local coordinate system (C, r]) is defined as centered at (a, 1 — a) so that the equations of motion are... 

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