One-dimensional three-body problem via symbolic dynamics

Abstract

Symbolic dynamics is applied to the one-dimensional three-body problem with equal masses. The sequence of binary collisions along an orbit is expressed as a symbol sequence of two symbols. Based on the time reversibility of the problem and numerical data, inadmissible (i.e., unrealizable) sequences of collisions are systematically found. A graph for the transitions among various regions in the Poincare section is constructed. This graph is used to find an infinite number of periodic sequences, which implies an infinity of periodic orbits other than those accompanying a simple periodic orbit called the Schubart orbit. Finally, under reasonable assumptions on inadmissible sequences, we prove that the set of admissible symbol sequences forms a Cantor set. (c) 2000 American Institute of Physics.

The classical Coulomb three-body problem in the collinear eZe configuration

“…As established for celestial problem (Tanikawa and Mikkola, 2000), the following properties are also true for our Coulomb system: Property 4.1: A trajectory in the (Q 1 , Q 2 )-plane transversely crosses the line Q 1 = Q 2 except at (θ, R) = (0, 0), if it does at all. Property 4.2: If a trajectory crosses the line Q 1 = Q 2 on the (Q 1 , Q 2 )-plane, a double collision occurs before the trajectory again crosses it.…”

“…Our numerical calculation up to the word length 15 showed that the inadmissible words are only the words including the words 1122 and 2211. Note that 1122 and 2211 are also inadmissible words for the celestial problem(the case of (Z, ξ) = (−1, 1)) and other inadmissible words exist (Tanikawa and Mikkola, 2000). On the other hand, for the case of the helium (Fig.6), i.e., large ξ, there is no torus.…”

“…If we set (Z, ξ) = (−1, 1), the system is equivalent to the system which was investigated in gravitational three-body problem (Tanikawa and Mikkola, 2000). We set the total momentum to be zero and change the variables q…”

“…The following two things are numerically checked for our Coulomb systems (i.e., the case of (Z, ξ) = (1, 1) and the case of the helium) as observed for celestial problem (Tanikawa and Mikkola, 2000): The plane D 1 is divided into two regions of the points z having Ξ 1 (χ (+) (z)) = 1 · 22 and 1 · 21. Furthermore, it divided into four regions of the points z having Ξ 2 (χ (+) (z)) = 1 · 222, 1 · 221, 1 · 212 and 1 · 211.…”

“…Parameters of these systems are the mass ratio ξ = m n /m e and the charge Z of nucleus, where m n (m e ) are the mass of nucleus(electron), respectively. We employ numerical computation for these systems with the aid of analytical tool(the triple collision manifold(TCM)) (McGehee, 1974) and symbolic dynamics (Tanikawa and Mikkola, 2000) from celestial mechanics. The TCM is a manifold which is a set of the initial or final conditions of the triple collision orbit, i.e., thus just the triple collision points.…”