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.

Symmetrical motion, Hill stable pair, Distal motion, Elliptic keplerian motion

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 [2-4]. 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 the general equations of motion of the three-body problem, giving them the desired shape already in the process of research.

So, in the study of symmetric motions, we will first of all rely on the basic equations of the three-body problem representing them in the form [7]:

ρ′′1 = μ2ρ2 - ρ1|ρ12|3 + μ3ρ3 - ρ1|ρ13|3,

ρ′′2 = - μ1ρ2 - ρ1|ρ12|3 + μ3ρ3 - ρ2|ρ23|3, (1.1)

ρ′′3 = - μ1ρ3 - ρ1|ρ13|3 - μ2ρ3 - ρ2|ρ23|3,

Where the prime symbol means differentiation with respect to τ(τ = tGM−−−−√/r320); μi = mi/M, M = m1 + m2 + m3;r0 is a parameter that has the dimension of the unit of length. In equations (1.1), we have ρi=ri/r0 ,i = 1,2,3, where ri are the radius vectors of points in the inertial reference frame with the origin at the center of mass of mi. The parameter ro having the dimension of the unit of length is included into the expression of τ in order to deal with dimensionless quantities since it is very convenient from the viewpoint of our subsequent study.

In what follows, we essentially use the conservative property of system (1.1), i.e. the existence of the energy integral

12∑i3μiρ′2i - ∑i < jμiμj|ρij| = h = const, (1.2)

and the vector integral of angular momentum

∑i3μ1(ρi × ρ′i) = C (1.3)

We assume that C ≠ 0.

Further, without loss of generality, we also assume that the equality

∑i3μiρi = 0 (1.4)

is satisfied; This means that the origin of the reference system is located at the center of mass of the material points (bodies).

As is shown in [6], if two masses are equal, then we arrive at the manifold of symmetric motions

ρ′′12 = - 2μρ12|ρ12|3 - μ3ρ12∣∣|ρ13|3∣∣, (2.1)

ρ′′3 = - ρ3|ρ13|3,

Where |ρ13| = |ρ23|,μ1 = μ2 = μ, 2μ + μ3 = 1, and the following equality is satisfied:

ρ23 = μ2(− ρ212 + 4ρ213) (2.2)

A distinctive feature of the manifold of symmetric motions of system (2.1) is the validity of equalities

ρ12 × ρ′12 = C1, ρ3 × ρ′3 = C2, (2.3)

Where C1,C2 are constant vectors. This enables us to perform a qualitative study of system (2.1) given that

ν212 = ρ′212 = ρ′212 + |ρ12 × ρ′12|2ρ212, (2.4)

ν23 = ρ′23 = ρ′23 + |ρ3 × ρ′|2ρ23, (2.5)

and reduce it to the following system with two degrees of freedom:

ρ212'' = 2ν212 - 4μρ12 - 2μ3ρ212ρ313, (2.6)

ρ23'' = 2ν23 - 2ρ23ρ313

If C2 = 0, then system (2.6) admits a motion, for which the body having the mass μ3 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:

ρ′12ρ13 - ρ12ρ′13 = ρ′12ρ23 - ρ12ρ′23 = ρ′13ρ23 - ρ13ρ′23 (2.7)

If we notice that in the case under consideration

ρ′12ρ13 - ρ12ρ′13 = ρ′12ρ23 - ρ12ρ′23 = ρ′13ρ23 - ρ13ρ′23 = 0 (2.8)

Which follows from equations (1.1) in the form

ρ′′12 = - (1 - μ3)ρ12|ρ12|3 + μ3(- ρ13|ρ13|3 + ρ23|ρ22|3),

ρ′′13 = - (1 - μ2)ρ13|ρ13|3 - μ2(- ρ12|ρ12|3 + ρ23|ρ23|3), (2.9)

ρ′′23 = - (1 - μ1)ρ23|ρ23|3 + μ1(ρ12|ρ12|3 - ρ13|ρ23|3),

and, in turn, the equalities

(ρ12ρ13)′ = ρ′12ρ13 + ρ12ρ′13 = 12(ρ212 + ρ213 - ρ223)′ = 12(ρ212)′,

(ρ12ρ23)′ = ρ′12ρ23 + ρ12ρ′23 = 12( - ρ212 + ρ213 - ρ223)′ = 12( - ρ212)′, (2.10)

(ρ13ρ23)′ = ρ′13ρ23 + ρ13ρ′23 = 12( - ρ212 + ρ213 - ρ223)′ = 12( - ρ212 + 2ρ213)′

are valid, then we obtain the manifold of symmetric motions in the form

ρ212'' = 2ν212 - 4μρ12 - 2μ3ρ212ρ313,

ρ213'' = 2ν213 - 2ρ13 + μ(- 1ρ12 + ρ212ρ313), (2.11)

E′12 = μ3ρ212′(1ρ312 - 1ρ313),

E′13 = - μ2ρ212′(1ρ312 - 1ρ313),

Where,

E12 = ν212 - 2ρ12, E13 = ν213 - 2ρ13 (2.12)

Taking into account equalities (2.2) - (2.5), as well as the equality

ν23 = − μ2ν212 + 4μ2ν213, (2.13)

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

μ2E12 + 2μμ3E13 = 2h (2.14)

Further, we restrict ourselves to such symmetric motions of system (2.11) (or (2.6)) that belong to the set

Ω = {(ρ,ρ′):T − U = h < 0} (2.15)

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 ρ12,ρ13,ρ3 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

ρ2′′12 = 2((ρ212′)24ρ212 + |C1|2ρ212) - 4μρ12 - 2μ3ρ212ρ313, (3.1)

ρ2′′13 = 2hμμ3 - μμ3((ρ212′)24ρ212 + |C1|2ρ212) + μ(2 - μ3)μ3ρ12 + 2ρ13 + μρ212ρ313

To determine the equilibrium positions, we arrive at the equations

2((ρ212′)24ρ212 + |C1|2ρ212)- 4μρ12 - 2μ3ρ212ρ313 = 0, (3.2)

2hμμ3 - μμ3((ρ212′)24ρ212 + |C1|2ρ212) + μ(2 - μ3)μ3ρ12 + 2ρ13 + μρ212ρ313 = 0

Given the fact that in the equilibrium position ρ212′ = 0, we rewrite Eqs. (3.2) in the form

2(c2ρ412) - 4μρ312 - 2μ31ρ313 = 0, (3.3)

2hμμ31ρ212 - μμ3(c2ρ412) + μ(2 - μ3)μ31ρ312 + 1ρ2122ρ13 + μ1ρ313 = 0,

Where, c2 = |C1|2. Thus, we have a system of two nonlinear algebraic equations for the variables 1/ρ12 and 1/ρ13. It is required to prove that this system has at least one positive solution, given that 1/ρ12 and 1/ρ13 are positive values in their meaning.

Solving the first equation of system (3.3) with respect to 1/ρ313 and substituting the resulting expression in the second equation of this system, we obtain

1ρ212[2hμμ3 + μμ31ρ12 + 2ρ13] = 0 (3.4)

Since in the case under consideration, 1/ρ12 is positive, then on the basis of (3.4) we have

1ρ13 = - 1μ3[hμ + μ2 1ρ12], (3.5)

hence

1ρ12 < - 2hμ2 (3.6)

At the same time, on the basis of the first equation of system (3.3), we have

1ρ12 > 2μc2 (3.7)

From the compatibility condition for inequalities (3.6) and (3.7) we obtain

hc2 + μ3 < 0 (3.8)

Now we substitute the value 1/ρ13, which is expressed by the right-hand side of equality (3.5), into the first equation of system (3.3). As a result, we arrive at the equation

8c2μ3μ23x4 + μ4(μ2 - 16μ23)x3 + 6μ4hx2 + 12μ2h2x + 8h3 = 0, (3.9)

Where, x = 1/ρ12.

Consider the left side of equation (3.9), which we denote by P4(x), for x =2μ/c2 and

x = − 2h/μ2,

respectively. As a result, we get

P4(2μc2) = 8c6(hc2 + μ3)3, (3.10)

P4(− 2hμ2) = 128μ23h3μ5(hc2 + μ3), (3.11)

As can be seen from equalities (3.10) and (3.11), when passing from the value x =2μ/c2 to the value x = − 2h/μ2, the polynomial P4(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).

Thus, equalities (3.6), (3.7), (3.10) and (3.11) allow us to obtain the following

Assertion 1. The manifold of symmetric motions (2.11) (or (2.6)) admits stationary motions (for h < 0 and C2 ≠ 0), if and only if the condition is satisfied:

hc2 + μ3 < 0

Necessity, as we could see above, follows from inequalities (3.6) and (3.7), sufficiency follows from equalities (3.10) and (3.11).

Since the distance ρ3 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 μ3 there is also its rotational motion when C2 ≠ 0.

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 ρ12 and r instead of ρ12 respectively.

Let us write down the equation for r in the form

r2′′ = 2ν2 − 2r, (4.1)

Where,

ν2 = r′2 = r′2 + |r × r′|2r2 = r′2 + c∗2r2 (4.2)

The energy integral is represented as

μ2(r′2 − 2r) = 2h˜ = const (4.3)

Next we use the known equalities for the two-body problem

1r = 1c∗2(1 + ecos f), (4.4)

r′2 = e2c∗2sin2f, (4.5)

ν2 = 1c∗2[1 + e2 + 2ecos f], (4.6)

Where the constant

e = μ2 + 2c∗2h˜μ2−−−−−−−−−−√ (4.7)

is the eccentricity of the elliptical orbit, f is a true anomaly.

We recall some key definitions that we will use below.

Definition 1. We say that the motion ρ(τ) = (ρ1,ρ2,ρ3)τ of system (1.1) is distal if the following inequality is satisfied:

|ρij(τ)| ≥ c1 ∀τ∈R, ∀i < j, 0 < c1 = const

Definition 2. In accordance with [10], we say that a fixed pair of mass points (μi,μj),i < j, of system (1.1) is Hill stable if the following inequality is satisfied:

|ρij(τ)| < c2 ∀τ∈R, 0 < c2 = const

According to [6], if h < 0 and |C1| ≠ 0, then the symmetric motions belonging to the manifold (2.1) are distal and, in addition, the pair of material points (μ,μ) is Hill stable.

Assertion 2. Let ρ(τ) = (ρ1,ρ2,ρ3)τ be a symmetric motion of system (2.1) belonging to the set Ω.

Then, if

μ3 + c2h ≤ 0, (4.8)

Where

c2 = |ρ12 × ρ′12|2, c = const, (4.9)

then the symmetric motion is bounded.

Proof. Further, the energy integral is conveniently written as

12μ(μ3ρ′23 + μ2ρ′213) - 2μ2ρ12 - 4μμ3ρ13 = 2h (4.10)

Rewriting it in the form

μ2[12μ(ρ′212 + |ρ12 × ρ′12|2ρ212) - 2ρ12] + μ32μρ′23 - 4μμ3ρ13 = 2h, (4.11)

We consider it as a quadratic equation with respect to the value 1/ρ12. As a result, we obtain

c2μ2(1ρ12)2 − 2μ21ρ12 + (μρ′2122 + μ32μρ′23 − 4μμ3ρ13 − 2h) = 0 (4.12)

In view of (4.12), we have

1ρ12 = 2μc2 ± 2c2μ−−√μ3 + c2h − c22(μρ′2122 + μ32μρ′23 − 4μμ3ρ13)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−⎷ (4.13)

and the resulting equality is represented as

1ρ12 − 2μc2 = ± 2c2μ−−√μ3 + c2h − c22(μρ′2122 + μ32μρ′23 − 4μμ3ρ13)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−⎷ (4.14)

Under the conditions of the assertion 2, the left-hand side of equality (4.14) is always a real number.

Suppose now that the considered symmetric motion ρ(τ) = (ρ1,ρ2,ρ3)τ is not bounded under the conditions of the assertion. Then there exists a sequence {τk},k = 1,2,3,…, such that

limk→∞τk = ∞, limk→∞¯¯¯¯¯¯ ρ13(τk) = ∞, ρ13(τk) = |ρ13(τk)|. (4.15)

We first consider the case of condition (4.8), when the strict inequality

μ3 + c2h < 0

holds. Then, in accordance with (4.15), for the sequence {τk}, there exists a sufficiently large number k such that 1/ρ13(τk) becomes an arbitrarily small number, and, as a result, the right-hand side of equality (4.14) becomes imaginary. We get a contradiction.

Let now

μ3 + c2h = 0

Then, if we assume that motion is unbounded in this case, then (4.15) holds, and, consequently, in accordance with equations (2.1), ρ12(τ) approaches an elliptic Keplerian motion as k → ∞ And then, taking into account (3.5), we see that the term ρ′212 under the sign of radical in (4.14), which is associated with the pair (μ,μ), is representable in the following form:

ρ′212 = e˜2c˜2sin2f˜, (4.16)

Where e˜,c˜ and f˜ = f˜(τ) have the same meaning as e,c* and f in equality (3.5).

Since for the elements of the sequence {τk} we have

1ρ13(τk) → 0, (4.17)

as k → ∞, then the limit expression for the sum of terms under the sign of radical in (4.14) takes the form

{− c22(μρ′2122 + μ32μρ′23 − 4μμ3ρ13)}∞ = − c2μ4{e˜2c˜2sin2f˜ + μ3μ2ρ′23} (4.18)

We now consider equality (4.18) in more detail. The function sin2f˜ in its right-hand side is equal to one for f˜ = (2i + 1)π/2, i = 0,1,2,…, and since we study the motions belonging to the set Ω, the right-hand side of equality (4.18) becomes negative for sin2f˜ = 1.

As noted above, the pair of material points (μ,μ),is Hill stable on manifold (2.1) and the motion in question is distal, which makes the velocities of the material points of system (2.1) limited. The period of the elliptic Keplerian motion approaching ρ12(τk) as k → ∞ is finite. Within this period, the function sin2f˜, being continuous, takes all its values. Thus, taking into account (4.17) and (4.18), we get every reason to assert that there exists a value τ∗(k) such that

[− c22(μρ′2122 + μ32μρ′23 - 4μμ3ρ13)]τ = τ∗(k)

becomes negative and, as a result, the right-hand side of equality (4.14) becomes imaginary. As we noted above, under the conditions of the assertion 2, the left-hand side of equality (4.14) is always valid. We arrive at a contradiction, whence we conclude that the assertion 2 is true.

As it is implied by the scheme of proof of assertion 2, boundedness of symmetrical motions remains to be true also in the case when the constant μ3 + c2h is positive, but is quite small.

So, if the oscillating motions exist, then they can be realized only if the condition: μ3 + c2h > δ is satisfied, where δ is a small positive number. This fact may be of some practical interest.

In our proof of the assertion 2 on boundedness of symmetric motions, a key point is the fact that in case of moving the third body to infinity, under the conditions of the assertion 2, the pair (μ,μ) acquires the properties of an elliptic Keplerian motion. Moreover, as follows from the proofs of the assertions 1 and 2, the boundedness of symmetric motions is essentially associated with the absolute value of the angular momentum of the pair (μ,μ) and the constant of the energy integral h , as well as with their relation.