On an Application of the Hill Approach to the General Case of the Three-body Problem

Now, in the era of active space exploration, the three-body problem has quite rightly grown from a purely academic problem to an applied one. Computer modeling, analytical methods, and numerical methods (the development of which was largely stimulated by the three-body problem itself) have made it possible to achieve noticeable success in its research (Marchal 1990; Arnold et al. 2002; Celletti 2010; Georgakarakos 2008). Nevertheless, the problem remains unsolved in full to this day, and the qualitative study of the motion in the system is still relevant. Within the framework of this study, it is still important, both from theoretical and practical viewpoints, to find the conditions under which three bodies remain in a bounded region of Euclidean space. Below, based on Hill's approach to the circular restricted three-body problem, we indicate sufficient conditions that ensure the bounded motion in the case of the general three-body problem.

As basic equations, we consider the three-body problem in the form (Sosnitskii 2008):

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{\rho }}}_{1}^{{\prime\prime} } & = & {\mu }_{2}\displaystyle \frac{{{\boldsymbol{\rho }}}_{2}-{{\boldsymbol{\rho }}}_{1}}{| {{\boldsymbol{\rho }}}_{12}{| }^{3}}+{\mu }_{3}\displaystyle \frac{{{\boldsymbol{\rho }}}_{3}-{{\boldsymbol{\rho }}}_{1}}{| {{\boldsymbol{\rho }}}_{13}{| }^{3}},\\ {{\boldsymbol{\rho }}}_{2}^{{\prime\prime} } & = & -{\mu }_{1}\displaystyle \frac{{{\boldsymbol{\rho }}}_{2}-{{\boldsymbol{\rho }}}_{1}}{| {{\boldsymbol{\rho }}}_{12}{| }^{3}}+{\mu }_{3}\displaystyle \frac{{{\boldsymbol{\rho }}}_{3}-{{\boldsymbol{\rho }}}_{2}}{| {{\boldsymbol{\rho }}}_{23}{| }^{3}},\\ {{\boldsymbol{\rho }}}_{3}^{{\prime\prime} } & = & -{\mu }_{1}\displaystyle \frac{{{\boldsymbol{\rho }}}_{3}-{{\boldsymbol{\rho }}}_{1}}{| {{\boldsymbol{\rho }}}_{13}{| }^{3}}-{\mu }_{2}\displaystyle \frac{{{\boldsymbol{\rho }}}_{3}-{{\boldsymbol{\rho }}}_{2}}{| {{\boldsymbol{\rho }}}_{23}{| }^{3}},\end{array}\end{eqnarray} \tag{ 1 }$$

where the prime sign means differentiation with respect to τ ($\tau =t\sqrt{{GM}}/{r}_{0}^{3/2}$), μ_i = m_i /M, M = m₁ + m₂ + m₃, r₀ is a parameter having the dimension of a unit of length. In Equation (1), ρ _i = r _i /r₀ (i = 1, 2, 3), where r _i are radius vectors of points in an inertial reference system with the origin at the center of mass for m_i . The parameter r₀ with the dimension of a unit length is introduced in order to subsequently work with dimensionless quantities, which is very convenient for our presentation.

If we set μ₃ = 0 in Equations (1), then we arrive at the restricted three-body problem:

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{\rho }}}_{1}^{{\prime\prime} } & = & \mu \displaystyle \frac{{{\boldsymbol{\rho }}}_{12}}{| {{\boldsymbol{\rho }}}_{12}{| }^{3}},\\ {{\boldsymbol{\rho }}}_{2}^{{\prime\prime} } & = & -(1-\mu )\displaystyle \frac{{{\boldsymbol{\rho }}}_{12}}{| {{\boldsymbol{\rho }}}_{12}{| }^{3}},\\ {{\boldsymbol{\rho }}}_{3}^{{\prime\prime} } & = & -(1-\mu )\displaystyle \frac{{{\boldsymbol{\rho }}}_{13}}{| {{\boldsymbol{\rho }}}_{13}{| }^{3}}-\mu \displaystyle \frac{{{\boldsymbol{\rho }}}_{23}}{| {{\boldsymbol{\rho }}}_{23}{| }^{3}},\end{array}\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray*}&&\mu =\displaystyle \frac{{m}_{2}}{{m}_{1}+{m}_{2}},\quad 0\lt \mu \leqslant \displaystyle \frac{1}{2}.\end{eqnarray*}$$

Since

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{\rho }}}_{13} & = & \displaystyle \frac{1}{{\mu }_{1}+{\mu }_{2}}({{\boldsymbol{\rho }}}_{3}+{\mu }_{2}{{\boldsymbol{\rho }}}_{12}),\\ {{\boldsymbol{\rho }}}_{23} & = & \displaystyle \frac{1}{{\mu }_{1}+{\mu }_{2}}({{\boldsymbol{\rho }}}_{3}-{\mu }_{1}{{\boldsymbol{\rho }}}_{12})\end{array}\end{eqnarray} \tag{ 3 }$$

and for the restricted three-body problem,

$$\begin{eqnarray}&&{{\boldsymbol{\rho }}}_{13}=({{\boldsymbol{\rho }}}_{3}+\mu {{\boldsymbol{\rho }}}_{12}),\quad {{\boldsymbol{\rho }}}_{23}=[{{\boldsymbol{\rho }}}_{3}-(1-\mu ){{\boldsymbol{\rho }}}_{12}],\end{eqnarray} \tag{ 4 }$$

it is also useful to consider the following representations of systems (1) and (2):

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{\rho }}}_{12}^{{\prime\prime} } & = & -({\mu }_{1}+{\mu }_{2})\displaystyle \frac{{{\boldsymbol{\rho }}}_{12}}{| {{\boldsymbol{\rho }}}_{12}{| }^{3}}+{\mu }_{3}\left(-\displaystyle \frac{{{\boldsymbol{\rho }}}_{13}}{| {{\boldsymbol{\rho }}}_{13}{| }^{3}}+\displaystyle \frac{{{\boldsymbol{\rho }}}_{23}}{| {{\boldsymbol{\rho }}}_{23}{| }^{3}}\right),\\ {{\boldsymbol{\rho }}}_{3}^{{\prime\prime} } & = & -{\mu }_{1}\displaystyle \frac{{{\boldsymbol{\rho }}}_{13}}{| {{\boldsymbol{\rho }}}_{13}{| }^{3}}-{\mu }_{2}\displaystyle \frac{{{\boldsymbol{\rho }}}_{23}}{| {{\boldsymbol{\rho }}}_{23}{| }^{3}},\end{array}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{12}^{{\prime\prime} } & = & -\displaystyle \frac{{{\boldsymbol{\rho }}}_{12}}{| {{\boldsymbol{\rho }}}_{12}{| }^{3}},\\ {{\boldsymbol{\rho }}}_{3}^{{\prime\prime} } & = & -(1-\mu )\displaystyle \frac{{{\boldsymbol{\rho }}}_{13}}{| {{\boldsymbol{\rho }}}_{13}{| }^{3}}-\mu \displaystyle \frac{{{\boldsymbol{\rho }}}_{23}}{| {{\boldsymbol{\rho }}}_{23}{| }^{3}}.\end{array}\end{eqnarray} \tag{ 6 }$$

Although it is a rather simplified model of the motion of three bodies (Szebehely 1967; Roy 1978), nevertheless, the circular restricted problem of three bodies (material points) finds many interesting applications even today (Makó & Szenkovits 2008; Szebehely 1967; Lian et al. 2013; Georgakarakos 2008; Qi & Xu 2014; Song et al. 2016; Salazar et al. 2017).

As Jacobi showed, it admits an integral

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{{{\boldsymbol{\rho }}}_{3}^{{\prime} }}^{2}-({{\boldsymbol{\rho }}}_{3}\times {{\boldsymbol{\rho }}}_{3}^{{\prime} }){| }_{\zeta }-\left(\displaystyle \frac{1-\mu }{{\rho }_{13}}+\displaystyle \frac{\mu }{{\rho }_{23}}\right)={h}^{* }={\rm{const}},\end{eqnarray} \tag{ 7 }$$

where ρ₁₃ = ∣ ρ ₁₃∣, ρ₂₃ = ∣ ρ ₂₃∣ and the expression $({{\boldsymbol{\rho }}}_{3}\times {{\boldsymbol{\rho }}}_{3}^{{\prime} }){| }_{\zeta }$ on the left side of equality (7) is the projection of the angular momentum of the small particle onto the axis O ζ in an inertial reference frame O ξ η ζ. The O ζ axis is perpendicular to the O ξ η plane in which two massive bodies move. As we see, the Jacobi integral is the sum of the energy of the small particle and the projection of its angular momentum onto the O ζ axis.

The existence of the Jacobi integral in the circular restricted three-body problem allowed Hill (Hill 1878) to prove the existence of bounded motions of the small particle in the case where the level constant h* of the Jacobi integral is negative and ∣h*∣ exceeds some critical value $\widetilde{{h}^{* }}\gt 0$. In what follows, this condition will be referred to as the Hill condition. If it holds, then the region of possible motions of the infinitesimal particle is the union of the region ω_H of bounded motions in coordinates (Hill region) and the region ω_nc of bounded motions in velocities, i.e., ω = ω_H ∪ ω_nc , and ω_H ∩ ω_nc = Ø. A distinctive feature of the region ω_nc is that it is free from collisions of bodies (material points). This explains the choice of the nc index to denote this region. Unlike ω_H , the region ω_nc is not bounded, which raises the problem of bounded motion of the small particle in ω_nc . However, relatively recently (Sosnitskii 2018), we managed to prove the boundedness of motions of the small particle belonging to ω_nc .

The structure of the region ω motivates the definitions used below, which relate to both the restricted three-body problem and the general one.

Definition 1. A fixed pair of points $({\mu }_{i},{\mu }_{j}),i\lt j,$ of system (1) is called to be Lagrange stable if the following inequality is satisfied:

$$\begin{eqnarray}&&{c}_{1}\leqslant | {{\boldsymbol{\rho }}}_{{ij}}(\tau )| \leqslant {c}_{2}\quad \forall \tau \in R=]-\infty ,\infty [,\end{eqnarray} \tag{ 8 }$$

where ${c}_{1},{c}_{2}$ are positive constants.

Definition 2. A movement ${\boldsymbol{\rho }}(\tau )={\left({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{{\boldsymbol{\rho }}}_{3}\right)}^{T}$ of system (1) is called to be Lagrange stable if the following condition is satisfied:

$$\begin{eqnarray*}&&{c}_{3}\leqslant | {{\boldsymbol{\rho }}}_{{ij}}(\tau )| \leqslant {c}_{4}\quad \forall \tau \in R,\quad \forall i\lt j,\end{eqnarray*}$$

where ${c}_{3},{c}_{4}$ are positive constants.

Definition 3. We say that the motion ${\boldsymbol{\rho }}{(\tau )=({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{{\boldsymbol{\rho }}}_{3})}^{T}$ of system (1) is Hill stable if the following condition is satisfied:

$$\begin{eqnarray*}&&| {{\boldsymbol{\rho }}}_{{ij}}(\tau )| \leqslant {c}_{5}\quad \forall \tau \in R,\quad \forall i\lt j,\end{eqnarray*}$$

where c₅ is positive constant.

Definition 4. We say that the motion ${\boldsymbol{\rho }}{(\tau )=({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{{\boldsymbol{\rho }}}_{3})}^{T}$ of system (1) is distal if the following inequality is satisfied:

$$\begin{eqnarray*}&&| {{\boldsymbol{\rho }}}_{{ij}}(\tau )| \geqslant {c}_{6}\quad \forall \tau \in R,\quad \forall i\lt j,\quad 0\lt {c}_{6}={\rm{const}}.\end{eqnarray*}$$

Remark 1. Since the quantities ${{\boldsymbol{\rho }}}_{i}$ in Equations (1) are relative lengths, without the loss of generality we can further assume in Equation (8) that ${c}_{1}\geqslant 1$.

In the inertial reference frame O ξ η ζ the Jacobi integral takes the form

$$\begin{eqnarray}&&\begin{array}{l}{{{\boldsymbol{\rho }}}_{3}^{{\prime} }}^{2}-2(\xi \eta ^{\prime} -\eta \xi ^{\prime} )-2\left(\displaystyle \frac{1-\mu }{{\rho }_{13}}+\displaystyle \frac{\mu }{{\rho }_{23}}\right)\\ \,=\,2{h}^{* },\quad {h}^{* }={\rm{const}},\end{array}\end{eqnarray} \tag{ 9 }$$

where ${{{\boldsymbol{\rho }}}_{3}^{{\prime} }}^{2}=\xi {{\prime} }^{2}+\eta {{\prime} }^{2}+\zeta {{\prime} }^{2}$. However, if we use a coordinate system that rotates with unit angular velocity around the O ζ axis, i.e.,

$$\begin{eqnarray*}&&\xi =x\cos \tau -y\sin \tau ,\end{eqnarray*}$$

$$\begin{eqnarray*}&&\eta =x\sin \tau +y\cos \tau ,\end{eqnarray*}$$

where (x, y, z) are coordinates of the small particle relative to the coordinate system that rotates, the Jacobi integral turns into one of the equalities

$$\begin{eqnarray}&&{{{\boldsymbol{\rho }}}_{3}^{{\prime} }}^{2}-2({xy}^{\prime} -{yx}^{\prime} )-2\left({x}^{2}+{y}^{2}+\displaystyle \frac{1-\mu }{{\rho }_{13}}+\displaystyle \frac{\mu }{{\rho }_{23}}\right)=2{h}^{* },\end{eqnarray} \tag{ 10 }$$

or

$$\begin{eqnarray}&&x{{\prime} }^{2}+y{{\prime} }^{2}+z{{\prime} }^{2}-\left({x}^{2}+{y}^{2}\right)-\displaystyle \frac{2(1-\mu )}{{\rho }_{13}}-\displaystyle \frac{2\mu }{{\rho }_{23}}=2{h}^{* }.\end{eqnarray} \tag{ 11 }$$

In this case, the second vector equation of system (6) takes the following form (see Szebehely 1967):

$$\begin{eqnarray}\begin{array}{rcl}x^{\prime\prime} -2y^{\prime} & = & x-(1-\mu )\displaystyle \frac{x-\mu }{{\rho }_{13}^{3}}-\mu \displaystyle \frac{x+1-\mu }{{\rho }_{23}^{3}},\\ y^{\prime\prime} +2x^{\prime} & = & y-(1-\mu )\displaystyle \frac{y}{{\rho }_{13}^{3}}-\mu \displaystyle \frac{y}{{\rho }_{23}^{3}},\\ z^{\prime\prime} & = & -(1-\mu )\displaystyle \frac{z}{{\rho }_{13}^{3}}-\mu \displaystyle \frac{z}{{\rho }_{23}^{3}},\end{array}\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{13}^{2} & = & {\left(x-\mu \right)}^{2}+{y}^{2}+{z}^{2},\\ {\rho }_{23}^{2} & = & {\left(x+1-\mu \right)}^{2}+{y}^{2}+{z}^{2}.\end{array}\end{eqnarray} \tag{ 13 }$$

It was the Jacobi integral in the form (11) that Hill used. Within the framework of Hill's approach, for the existence of the regions ω_H and ω_nc , the determining factor is the inequality

$$\begin{eqnarray}&&({x}^{2}+{y}^{2})+2\left(\displaystyle \frac{1-\mu }{{\rho }_{13}}+\displaystyle \frac{\mu }{{\rho }_{23}}\right)+2{h}^{* }\geqslant 0,\quad \end{eqnarray} \tag{ 14 }$$

in which the controlled parameter is the level constant h^* of the Jacobi integral. As follows from the above definitions, the pair of massive bodies, which is located in the O ξ η plane, is Lagrange stable, and the motions of the small particle belonging to the regions ω_H and ω_nc are, respectively, Hill stable and Lagrange stable.

As we mentioned above, the Jacobi integral, at least in the inertial coordinate system, is the sum of the energy of the small particle and the projection of its angular momentum onto the O ζ axis. At the same time, it is worth noting that the region ω_nc is precisely generated by this projection, which directly follows from Equation (9). If we take into account that the general three-body problem allows both an energy integral and an angular momentum integral, then the question arises whether it is possible to use Hill's approach to obtain conditions for bounded motions in the case of the general three-body problem, at least under additional restrictions. As we will show below, if by analogy with the circular restricted problem we assume the presence of a Lagrange-stable pair in the general three-body problem, then the answer to this question is positive.

In a qualitative study of motion in the case of the three-body problem, the key role is played by the energy integral and the vector integral of the angular momentum, so we are going to use them below. In particular, in accordance with systems (1) and (5), we present the energy integral in the form

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\displaystyle \sum _{i}^{3}{\mu }_{i}{{\boldsymbol{\rho }}}_{i}{{\prime} }^{2}-\displaystyle \sum _{i\lt j}\displaystyle \frac{{\mu }_{i}{\mu }_{j}}{{\rho }_{{ij}}}=h={\rm{const}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\left(\displaystyle \frac{{\mu }_{1}{\mu }_{2}}{{\mu }_{1}+{\mu }_{2}}{{\boldsymbol{\rho }}}_{12}^{{\prime} 2}+\displaystyle \frac{{\mu }_{3}}{{\mu }_{1}+{\mu }_{2}}{{\boldsymbol{\rho }}}_{3}^{{\prime} 2}\right)-\displaystyle \sum _{i\lt j}\displaystyle \frac{{\mu }_{i}{\mu }_{j}}{{\rho }_{{ij}}}=h={\rm{const}}.\end{eqnarray} \tag{ 16 }$$

Here ρ_ij = ∣ ρ _ij ∣, i, j = 1, 2, 3. Further we restrict ourselves to the case where h < 0.

Along with the usual form of the vector integral of angular momentum

$$\begin{eqnarray}&&\displaystyle \sum _{i}^{3}{\mu }_{i}({{\boldsymbol{\rho }}}_{i}\times {{\boldsymbol{\rho }}}_{i}^{\prime} )={\boldsymbol{C}},\end{eqnarray} \tag{ 17 }$$

we also present it in the following form (Sosnitskii 2010):

$$\begin{eqnarray}&&{\mu }_{1}{\mu }_{2}({{\boldsymbol{\rho }}}_{12}\times {{\boldsymbol{\rho }}}_{12}^{{\prime} })+{\mu }_{3}({{\boldsymbol{\rho }}}_{3}\times {{\boldsymbol{\rho }}}_{3}^{{\prime} })={\boldsymbol{C}},\end{eqnarray} \tag{ 18 }$$

which corresponds to the equations of motion (5). We assume that C ≠ 0.

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

$$\begin{eqnarray}&&\displaystyle \sum _{i}^{3}{\mu }_{i}{{\boldsymbol{\rho }}}_{i}={\bf{0}}\end{eqnarray} \tag{ 19 }$$

holds, which fixes the origin of the reference system at the center of mass of the material points under consideration.

In our study, we use the above presented forms Equations (16) and (18) of integrals of energy and angular momentum.

Theorem 1. Let a motion ${\boldsymbol{\rho }}{(\tau )=({{\boldsymbol{\rho }}}_{12},{{\boldsymbol{\rho }}}_{3})}^{T}$ of system (5), which belongs to the set

$$\begin{eqnarray*}&&{\rm{\Omega }}=\left\{({\boldsymbol{\rho }}^{\prime} ,{\boldsymbol{\rho }}):T-U=h\lt 0\right\},\end{eqnarray*}$$

be associated with a Lagrange stable pair $({\mu }_{1},{\mu }_{2})$ $({\mu }_{1}\gt {\mu }_{2})$. If, in addition, the motion under consideration corresponds to a sufficiently large constant $(h-C)$ and a sufficiently small mass ${\mu }_{2}$, then it is either Hill stable or Lagrange stable.

Proof. Let us associate the axis $O\zeta$ of the inertial reference frame $O\xi \eta \zeta$ with the direction of the angular momentum ${\boldsymbol{C}}$. Then the projection of equality (18) onto the $O\zeta$ axis takes the form

$$\begin{eqnarray}&&{\mu }_{1}{\mu }_{2}({v}_{12\xi }{\rho }_{12\eta }-{\rho }_{12\xi }{v}_{12\eta })+{\mu }_{3}({v}_{3\xi }{\rho }_{3\eta }-{\rho }_{3\xi }{v}_{3\eta })=C,\end{eqnarray} \tag{ 20 }$$

where

$$\begin{eqnarray*}\begin{array}{rcl}{\rho }_{12\xi } & = & {{\boldsymbol{\rho }}}_{12}{| }_{\xi },\quad {\rho }_{12\eta }={{\boldsymbol{\rho }}}_{12}{| }_{\eta },\\ {v}_{12\xi } & = & {{\boldsymbol{\rho }}}_{12}^{{\prime} }{| }_{\xi },\quad {v}_{12\eta }={{\boldsymbol{\rho }}}_{12}^{{\prime} }{| }_{\eta },\\ {\rho }_{3\xi } & = & {{\boldsymbol{\rho }}}_{3}{| }_{\xi },\quad {\rho }_{3\eta }={{\boldsymbol{\rho }}}_{3}{| }_{\eta },\\ {v}_{3\xi } & = & {{\boldsymbol{\rho }}}_{3}^{{\prime} }{| }_{\xi },\quad {v}_{3\eta }={{\boldsymbol{\rho }}}_{3}^{{\prime} }{| }_{\eta },\quad C=| {\boldsymbol{C}}| .\end{array}\end{eqnarray*}$$

Let us rewrite equality (16) as follows:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\mu }_{1}{\mu }_{2}}{{\mu }_{1}+{\mu }_{2}}\left({v}_{12\xi }^{2}+{v}_{12\eta }^{2}+{v}_{12\zeta }^{2}\right)+\displaystyle \frac{{\mu }_{3}}{{\mu }_{1}+{\mu }_{2}}\left({v}_{3\xi }^{2}+{v}_{3\eta }^{2}+{v}_{3\zeta }^{2}\right)\\ \,-2\displaystyle \sum _{i\lt j}\displaystyle \frac{{\mu }_{i}{\mu }_{j}}{{\rho }_{{ij}}}=2h={\rm{const}}.\end{array}\end{eqnarray} \tag{ 21 }$$

Multiplying it by $({\mu }_{1}+{\mu }_{2})$ and subtracting from it equality (20) multiplied by two, we arrive at the equality

$$\begin{eqnarray}&&\begin{array}{l}{\mu }_{1}{\mu }_{2}[{\left({v}_{12\xi }-{\rho }_{12\eta }\right)}^{2}+{\left({v}_{12\eta }+{\rho }_{12\xi }\right)}^{2}+{v}_{12\zeta }^{2}]\\ \,+\,{\mu }_{3}[{\left({v}_{3\xi }-{\rho }_{3\eta }\right)}^{2}+{\left({v}_{3\eta }+{v}_{3\xi }\right)}^{2}+{v}_{3\zeta }^{2}]\\ \,=\,{\mu }_{1}{\mu }_{2}({\rho }_{12\xi }^{2}+{\rho }_{12\eta }^{2})+{\mu }_{3}({\rho }_{3\xi }^{2}+{\rho }_{3\eta }^{2})\\ \,+\,2({\mu }_{1}+{\mu }_{2})\displaystyle \sum _{i\lt j}\displaystyle \frac{{\mu }_{i}{\mu }_{j}}{| {{\boldsymbol{\rho }}}_{{ij}}| }+2({\mu }_{1}+{\mu }_{2})(h-C).\end{array}\end{eqnarray} \tag{ 22 }$$

Based on (22), we obtain the inequality

$$\begin{eqnarray}&&\begin{array}{l}{\mu }_{1}{\mu }_{2}({\rho }_{12\xi }^{2}+{\rho }_{12\eta }^{2})+{\mu }_{3}({\rho }_{3\xi }^{2}+{\rho }_{3\eta }^{2})\,+2({\mu }_{1}+{\mu }_{2})\\ \,\,\times \,\displaystyle \sum _{i\lt j}\displaystyle \frac{{\mu }_{i}{\mu }_{j}}{| {{\boldsymbol{\rho }}}_{{ij}}| }+2({\mu }_{1}+{\mu }_{2})(h-C)\geqslant 0,\end{array}\end{eqnarray} \tag{ 23 }$$

which, for convenience in our further study, can be rewritten in the form

$$\begin{eqnarray}&&\begin{array}{l}({\rho }_{3\xi }^{2}+{\rho }_{3\eta }^{2})+2({\mu }_{1}+{\mu }_{2})\left(\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}+\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\right)\\ \,\geqslant \,\displaystyle \frac{1}{{\mu }_{3}}\left[-2({\mu }_{1}+{\mu }_{2})(h-C)\right.\\ \,\left.-\,{\mu }_{1}{\mu }_{2}({\rho }_{12\xi }^{2}+{\rho }_{12\eta }^{2})-2({\mu }_{1}+{\mu }_{2})\displaystyle \frac{{\mu }_{1}{\mu }_{2}}{{\rho }_{12}}\right].\end{array}\end{eqnarray} \tag{ 24 }$$

From here, in particular, we conclude that the zero surface corresponding to the transformation of the left-hand side of equality (22) to zero has the form

$$\begin{eqnarray}&&\begin{array}{l}({\rho }_{3\xi }^{2}+{\rho }_{3\eta }^{2})+2({\mu }_{1}+{\mu }_{2})\left(\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}+\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\right)\\ \,=\,\displaystyle \frac{1}{{\mu }_{3}}\left\{-2({\mu }_{1}+{\mu }_{2})(h-C)-{\mu }_{1}{\mu }_{2}\right.\\ \,\left.\times \,\left[({\rho }_{12\xi }^{2}+{\rho }_{12\eta }^{2})+2\displaystyle \frac{({\mu }_{1}+{\mu }_{2})}{{\rho }_{12}}\right]\right\}.\end{array}\end{eqnarray} \tag{ 25 }$$

As we see, within the framework of the three-dimensional coordinate space related to ${{\boldsymbol{\rho }}}_{3}$, this surface is pulsating, and the smaller ${\mu }_{2}$, the less this pulsation. To verify this, in contrast to the circular restricted problem, where the zero surface has the form

$$\begin{eqnarray*}&&({x}^{2}+{y}^{2})+2\left(\displaystyle \frac{1-\mu }{{\rho }_{13}}+\displaystyle \frac{\mu }{{\rho }_{23}}\right)=-2{h}^{\ast }\geqslant 0,\end{eqnarray*}$$

in the present case it is enough to note that the right-hand side of equality (25) contains a variable term ${\mu }_{1}{\mu }_{2}[...]$ in addition to the constant term. This reflects the fact that the distance ${\rho }_{12}$ changes over time. Since the first term prevails over the second one according to the conditions of Theorem 1, it is quite natural to say in this case that the zero surface defined by Equation (25) is pulsating. Of course, it is true within the framework of the three-dimensional coordinate space generated by ${{\boldsymbol{\rho }}}_{3}$. Let us also recall here that for the original system of equations in the form (5), the dimension of the coordinate space is twice as large.

Due to the equality

$$\begin{eqnarray*}&&{\rho }_{3}^{2}={\rho }_{3\xi }^{2}+{\rho }_{3\eta }^{2}+{\rho }_{3\zeta }^{2},\end{eqnarray*}$$

even more so, based on Equation (24), the following inequality is true:

$$\begin{eqnarray}&&\begin{array}{l}{\rho }_{3}^{2}+2({\mu }_{1}+{\mu }_{2})\left(\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}+\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\right)\\ \,\geqslant \ \displaystyle \frac{1}{{\mu }_{3}}\left\{-2({\mu }_{1}+{\mu }_{2})(h-C)-{\mu }_{1}{\mu }_{2}\right.\\ \,\times \,\left.\left[({\rho }_{12\xi }^{2}+{\rho }_{12\eta }^{2})+2\displaystyle \frac{({\mu }_{1}+{\mu }_{2})}{{\rho }_{12}}\right]\right\}.\end{array}\end{eqnarray} \tag{ 26 }$$

Since

$$\begin{eqnarray}&&{\rho }_{3}^{2}=-{\mu }_{1}{\mu }_{2}{\rho }_{12}^{2}+{\mu }_{1}({\mu }_{1}+{\mu }_{2}){\rho }_{13}^{2}+{\mu }_{2}({\mu }_{1}+{\mu }_{2}){\rho }_{23}^{2},\end{eqnarray} \tag{ 27 }$$

Equation (26) can be represented in the form

$$\begin{eqnarray}&&\begin{array}{l}-{\mu }_{1}{\mu }_{2}{\rho }_{12}^{2}+{\mu }_{1}({\mu }_{1}+{\mu }_{2}){\rho }_{13}^{2}+{\mu }_{2}({\mu }_{1}+{\mu }_{2})\\ \ \ \ \times \,{\rho }_{23}^{2}+2({\mu }_{1}+{\mu }_{2})\left(\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}+\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\right)\\ \ \ \ \ \geqslant \,\displaystyle \frac{1}{{\mu }_{3}}\left\{-2({\mu }_{1}+{\mu }_{2})(h-C)\right.\\ \ \ \ \ -\,\left.{\mu }_{1}{\mu }_{2}\left[({\rho }_{12\xi }^{2}+{\rho }_{12\eta }^{2})\,+2\displaystyle \frac{({\mu }_{1}+{\mu }_{2})}{{\rho }_{12}}\right]\right\}.\end{array}\end{eqnarray} \tag{ 28 }$$

Taking into account that ${\rho }_{12}^{2}\geqslant ({\rho }_{12\xi }^{2}+{\rho }_{12\eta }^{2})$ and using Equation (28), we arrive at the inequality

$$\begin{eqnarray}&&\begin{array}{l}{\mu }_{1}({\mu }_{1}+{\mu }_{2}){\rho }_{13}^{2}+{\mu }_{2}({\mu }_{1}+{\mu }_{2}){\rho }_{23}^{2}\\ \ \ \ +\,2({\mu }_{1}+{\mu }_{2})\left(\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}+\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\right)\\ \ \ \ \ \geqslant \,\displaystyle \frac{({\mu }_{1}+{\mu }_{2})}{{\mu }_{3}}\left[-2(h-C)-{\mu }_{1}{\mu }_{2}\left({\rho }_{12}^{2}+\displaystyle \frac{2}{{\rho }_{12}}\right)\right],\end{array}\end{eqnarray} \tag{ 29 }$$

and this implies that

$$\begin{eqnarray}&&\begin{array}{l}{\mu }_{1}{\rho }_{13}^{2}\,+{\mu }_{2}{\rho }_{23}^{2}+2\left(\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}+\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\right)\\ \ \ \ \ \geqslant \,\displaystyle \frac{1}{{\mu }_{3}}\left[-2(h-C)-{\mu }_{1}{\mu }_{2}\left({\rho }_{12}^{2}+\displaystyle \frac{2}{{\rho }_{12}}\right)\right].\end{array}\end{eqnarray} \tag{ 30 }$$

If the conditions of the theorem are met, then according to Equation (30) either both distances ${\rho }_{13}$ and ${\rho }_{23}$ are very large, or one of the distances ${\rho }_{13}$ and ${\rho }_{23}$ is very small.

If ${\rho }_{13}$ or ${\rho }_{23}$ is very small, then due to Equation (30) for the region ω_3H of possible motions, we respectively obtain approximate inequalities

$$\begin{eqnarray}&&\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}\gtrsim \displaystyle \frac{1}{{\mu }_{3}}[-(h-C)],\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\gtrsim \displaystyle \frac{1}{{\mu }_{3}}[-(h-C)],\end{eqnarray} \tag{ 32 }$$

and thus the considered motion of Equation (5) is Hill stable. Here, the notation ω_3H corresponds to the fact that, under the conditions of Theorem 1, the region of possible motions is an analog of the Hill region in the circular restricted problem, and, as in Hill's case, we relate this region to the three-dimensional coordinate space generated by ${{\boldsymbol{\rho }}}_{3}$.

Now let both distances ${\rho }_{13}$ and ${\rho }_{23}$ be large. Taking into account (27), we rewrite inequality (30) in the form

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\rho }_{3}^{2}}{({\mu }_{1}+{\mu }_{2})}+2\left(\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}+\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\right)\\ \ \ \ \ \geqslant \,\displaystyle \frac{1}{{\mu }_{3}}\left[-2(h-C)-{\mu }_{1}{\mu }_{2}\left(\displaystyle \frac{{\rho }_{12}^{2}}{{\mu }_{1}+{\mu }_{2}}+\displaystyle \frac{2}{{\rho }_{12}}\right)\right],\end{array}\end{eqnarray} \tag{ 33 }$$

and, therefore, under the conditions of the theorem, we obtain the approximate inequality

$$\begin{eqnarray}&&{\rho }_{3}^{2}\gtrsim \displaystyle \frac{1}{{\mu }_{3}}[-2({\mu }_{1}+{\mu }_{2})(h-C)],\end{eqnarray} \tag{ 34 }$$

which determines the area of distal movements ${\omega }_{3{nc}}$. Again, the notation ${\omega }_{3{nc}}$ reflects the fact that the region ${\omega }_{3{nc}}$, which we associate with the three-dimensional coordinate space generated by ${{\boldsymbol{\rho }}}_{3}$, is free from collisions of bodies (material points) under the conditions of Theorem 1.

Due to Theorem 2 from Sosnitskii (2020), the motion under study belonging to ${\omega }_{3{nc}}$ is Lagrange stable. The proof of Theorem 1 is completed. $\square$

Remark 2. Despite all the similarities between the proposed approach and Hill's approach, there is a fundamental difference between the situations in the circular restricted problem and the general three-body problem. Theorem 1 above deals exclusively with a fixed motion accompanied by a Lagrange-stable pair. Unfortunately, the conditions of the theorem do not exclude the possibility that in the vicinity of this motion there may be both motions for which such a pair exists and motions for which such a pair does not exist. Under the conditions of the circular restricted three-body problem, all motions are always accompanied by the same Lagrange-stable pair, and therefore the regions ${\omega }_{H}$ and ${\omega }_{{nc}}$, in contrast to ω_3H and ${\omega }_{3{nc}}$, consist entirely of bounded and distal movements, respectively. In addition, in contrast to the circular restricted three-body problem, an essential requirement in the resulting theorem is smallness of one of the masses in the pair. This is a consequence of the fact that we consider the zero surface in a narrower three-dimensional configuration space compared to the dimension of the original system, which is associated with ${{\boldsymbol{\rho }}}_{3}$, making the influence of distances ${\rho }_{12}$ to estimates in Equations (31), (32), and (34) negligible because of the smallness of ${\mu }_{2}$. However, the relationship between the term $-2(h-C)$ and terms containing the factor ${\mu }_{2}$ is still more convenient to evaluate if we take into account the inequality in Equation (8). Taking Equation (8) into account, Equation (33) can be represented as

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\rho }_{3}^{2}}{({\mu }_{1}+{\mu }_{2})}+2\left(\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}+\displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\right)\\ \ \ \ \ \geqslant \,\displaystyle \frac{1}{{\mu }_{3}}\left[-2(h-C)-{\mu }_{1}{\mu }_{2}\left(\displaystyle \frac{{c}_{2}^{2}}{{\mu }_{1}+{\mu }_{2}}+\displaystyle \frac{2}{{c}_{1}}\right)\right].\end{array}\end{eqnarray} \tag{ 35 }$$

Although we obtain a rougher inequality in this case, however, unlike Equation (33), it contains only constant quantities in its right-hand side.

Remark 3. If we assume that we can measure the mutual distances ${\rho }_{{ij}}$ at any time, then, using Equation (30), we can obtain an upper estimate for the constant $-2(h-C)$.

Considering Equations (30) and (33) as key inequalities in the proof of Theorem 1, it is worth noting that their right-hand sides are divided by μ₃. Therefore, if we assume that μ₃ is small, then the absolute value of the right-hand sides can be significantly influenced not only by the constant −2(h − C) and the mass μ₂, but also by the mass μ₃. Thus, due to the smallness of μ₃, this makes it possible to weaken the influence of μ₂ on the absolute value of the right-hand sides of Equations (30) and (33). This fact is reflected in the following assertion.

Remark 4. Let a motion ${\boldsymbol{\rho }}{(\tau )=({{\boldsymbol{\rho }}}_{12},{{\boldsymbol{\rho }}}_{3})}^{T}$ of Equation (5), which belongs to the set

$$\begin{eqnarray*}&&{\rm{\Omega }}=\left\{({\boldsymbol{\rho }}^{\prime} ,{\boldsymbol{\rho }}):T-U=h\lt 0\right\},\end{eqnarray*}$$

be associated with a Lagrange stable pair $({\mu }_{1},{\mu }_{2})$ $({\mu }_{1}\gt {\mu }_{2})$. If, in addition, the motion under consideration corresponds to a sufficiently large constant $(h-C)$ and a sufficiently small mass ${\mu }_{3}$, then it is either Hill stable or Lagrange stable whenever the inequality

$$\begin{eqnarray*}&&\begin{array}{l}\left[-2(h-C)-{\mu }_{1}{\mu }_{2}\left(\displaystyle \frac{{\rho }_{12}^{2}}{{\mu }_{1}+{\mu }_{2}}+\displaystyle \frac{2}{{\rho }_{12}}\right)\right]\\ \ \ \ \ \geqslant \,c,\quad 0\lt c={\rm{const}}\end{array}\end{eqnarray*}$$

is satisfied.

To prove Theorem 2, we can use the previous scheme, but in this case in accordance with the conditions of the theorem, we have to use inequalities

$$\begin{eqnarray*}&&\displaystyle \frac{{\mu }_{1}}{{\rho }_{13}}\gtrsim \displaystyle \frac{1}{2{\mu }_{3}}c,\quad \displaystyle \frac{{\mu }_{2}}{{\rho }_{23}}\gtrsim \displaystyle \frac{1}{2{\mu }_{3}}c,\quad {\rho }_{3}^{2}\gtrsim \displaystyle \frac{1}{{\mu }_{3}}c\end{eqnarray*}$$

instead of the final inequalities in Equations (31), (32), and (34).

Analyzing the structure of the Jacobi integral in the circular restricted three-body problem, we propose an approach based on Hill's idea adapted to the general case of the three-body problem. As a result, we obtain theorems on the Lagrange stability and Hill stability in the general three-body problem. The theorems obtained can serve as additions to Theorem 2 from Sosnitskii (2020).