On Boundedness of Symmetrical Motions in the Three-Body Problem

In the present paper, we study the three-body problem in the case where two bodies have equal masses, which imply the existence of a manifold of symmetric motions. We find conditions of existence of bounded symmetric motions. These conditions can be useful for elucidating those key circumstances that cause the existence of oscillating final evolutions. For the analysis of boundedness of motions, both the structure of the manifold of symmetrical motions and the integrals of energy and angular momentum are essential.


Introduction
Symmetric motions in the three-body problem are usually associated with the problem of existence of oscillating eventual evolutions considered in the well-known work by Sitnikov [1]. Although that work is mainly devoted to the restricted elliptic three-body problem, it gave impetus to further research of the general three-body problem [2][3][4]. All the more, the possibility of existence of oscillating motions within the framework of the general three-body problem was admitted by Chazy [5].
In this paper, continuing author's research [6], we expand their spectrum somewhat.
Turning to the consideration of symmetric motions, we will not immediately write down the equations of motion that correspond to the manifold of symmetric motions, but we will start from In what follows, we essentially use the conservative property of system (1.1), i.e. the existence of the energy integral 3 2 and the vector integral of angular momentum ( ) We assume that 0 C ≠ .
Further, without loss of generality, we also assume that the equality is satisfied; This means that the origin of the reference system is located at the center of mass of the material points (bodies).

On the Manifold of Symmetric Motions of the Three-Body Problem
As is shown in [6], if two masses are equal, then we arrive at the manifold of symmetric motions If C 2 = 0, then system (2.6) admits a motion, for which the body having the mass 3 µ oscillates along an axis passing through the center of mass of the system and perpendicular to the plane of movement of the other two bodies with equal masses. It is the case that was considered by Sitnikov [1]. We arrived at system (2.6) based on equations (1.1). We now use the distance equations [8], taking into account the relations obtained in this article: If we notice that in the case under consideration 12 13 12 13 12 23 12 23 13 23 13 23 Which follows from equations (1.1) in the form (  )   13  23  12  12  3  3  3  3  3  12 13 22 23  12  13  2  2  3  3  3  13 12 23 13  12  23  1  1  3  3  3  23 12 23 are valid, then we obtain the manifold of symmetric motions in the form 3 12 3 3 12 13 12 13 13 12 13 Taking into account equalities (2.2) -(2.5), as well as the equality We see that this is also a system with two degrees of freedom. In particular, its first two equations form a closed system. The remaining two equations are its consequence.
The energy integral for system (2.11) has the form 2 12 3 13 Further, we restrict ourselves to such symmetric motions of system (2.11) (or (2.6)) that belong to the set

On Stationary Symmetric Motions
Based on the structure of the manifold of symmetric motions both in the form (2.6) and in the form (2.11), we see that the stationary symmetric motions, i.e. movements for which the distances 12 13 3 , , ρ ρ ρ are constant, correspond to the equilibrium positions of the systems of equations (2.6) or (2.11) respectively. Next, we dwell on the system (2.11). Taking into account the energy integral (2.14), we rewrite it in the form ( ) 2  2  2  2  12  1  2  12  12  3  2  2  3  12 12 12 13 2  2  2  12  1  3  2  12  13  2  2  3  3  3  12  12  3 12  13  13   C  2 -2  2  = - To determine the equilibrium positions, we arrive at the equations ( ) 2  2  2  2  12  1  12  3  2  2  3  12 12 12 13 . Thus, we have a system of two nonlinear algebraic equations for the variables 12 1 ρ and 13 1 ρ . It is required to prove that this system has at least one positive solution, given that Since in the case under consideration, 12 1 ρ is positive, then on the basis of (3.4) we have 13 3 12 From the compatibility condition for inequalities (3.6) and (3.7) we obtain 2 3 Now we substitute the value 13 1 ρ , which is expressed by the right-hand side of equality (3. As can be seen from equalities (3.10) and (3.11), when passing from the value the polynomial P 4 (x), changes sign. Therefore, equation (3.9) has a positive root, which, given (3.5), indicates the presence of a positive solution to system (3.3).
Since the distance 3 ρ in the case under consideration in accordance with (2.2) is constant, then, as we see, in addition to the oscillatory motion of a body with mass 3 µ there is also its rotational motion when

An Assertion on Boundedness of Symmetric Motions
For our further goals, we use some results on the two-body problem presented in [9]. Those results will be applied to the case where the masses of bodies are equal. In the framework of this case, we use r instead of 12 ρ and r instead of 12 ρ respectively.
Let us write down the equation for r in the form The energy integral is represented as Next we use the known equalities for the two-body problem is the eccentricity of the elliptical orbit, f is a true anomaly. We recall some key definitions that we will use below.
Suppose now that the considered symmetric motion ( ) ( We first consider the case of condition (4.8), when the strict inequality approaches an elliptic Keplerian motion as k → ∞ And then, taking into account (3.5), we see that the term 2