In general the libration points are the equilibria solutions of the dynamical system, which describes the motion of an infinitesimal body.

Since *Ω* = *Ω*(*x*, *y*), then
$$\frac{d\mathit{\Omega}}{dt}}=\dot{x}{\mathit{\Omega}}_{x}+\dot{y}{\mathit{\Omega}}_{y}.$$(19)

from Eqs. (13) and (19) the Jacobi integral can be written as
$${\dot{x}}^{2}+{\dot{y}}^{2}-2\mathit{\Omega}+C=0.$$(20)

The positions of the equilibrium points are the solutions of the following equations
$${\mathit{\Omega}}_{x}={\mathit{\Omega}}_{y}=0.$$(21)

where
$$\begin{array}{}{\mathit{\Omega}}_{x}& =(x-\mu )[{f}_{1}({r}_{1})+{q}_{1}(x,y,{r}_{1})]\\ & +(x-\mu +1)[{f}_{2}({r}_{2})+{q}_{2}(x,y,{r}_{2})],\\ {\mathit{\Omega}}_{y}& =y[{g}_{1}({r}_{1})+{g}_{2}({r}_{2})+{q}_{1}(x,y,{r}_{1})+{q}_{2}(x,y,{r}_{2})],\end{array}$$(22)

and
$$\begin{array}{}{f}_{i}({r}_{i})={\mu}_{i}\left[{n}^{2}-\left({\displaystyle \frac{1}{{r}_{i}^{3}}}+{\displaystyle \frac{3}{2{r}_{i}^{5}}}(2{A}_{2i}+4{A}_{1i}-{A}_{3i})\right)\right],\\ {g}_{i}({r}_{i})={\mu}_{i}\left[{n}^{2}-\left({\displaystyle \frac{1}{{r}_{i}^{3}}}+{\displaystyle \frac{3}{2{r}_{i}^{5}}}(4{A}_{2i}+2{A}_{1i}-{A}_{3i})\right)\right],\\ {q}_{i}(x,y,{r}_{i})={\displaystyle \frac{15{\mu}_{i}}{2{r}_{i}^{7}}}\left[{A}_{1i}{\left[x+(-1{)}^{i}{\mu}_{3-i}\right]}^{2}+{A}_{2i}{y}^{2}\right].\end{array}$$(23)

Since the principal axes are different in triaxial rigid bodies we can assume that the triaxial rigid body of mass *m*_{i}, *i* = 1, 2, be nearly a sphere with radius *R*_{0i} and thereby one obtains
$$\begin{array}{}{a}_{i}={R}_{0i}+{\sigma}_{1i},\\ {b}_{i}={R}_{0i}+{\sigma}_{2i},\\ {c}_{i}={R}_{0i}+{\sigma}_{3i},\end{array}$$(24)

where σ_{1i}, σ_{2i}, σ_{3i} ≪ 1. For investigations, see [5] and [25].

Substituting Eqs. (24) into Eqs. (12) one obtain
$$\begin{array}{}{A}_{1i}={\lambda}_{i}+{\delta}_{i}{\sigma}_{1i},\\ {A}_{2i}={\lambda}_{i}+{\delta}_{i}{\sigma}_{2i},\\ {A}_{3i}={\lambda}_{i}+{\delta}_{i}{\sigma}_{3i},\end{array}$$(25)

where
${\lambda}_{i}={\displaystyle \frac{{R}_{0i}^{2}}{5{R}^{2}}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\delta}_{i}={\displaystyle \frac{2{R}_{0i}}{5{R}^{2}}}.$

Substituting Eqs. (25) into Eqs. (22) and Eq. (18) with the help of Eqs. (23), we obtain
$$\begin{array}{}{\mathit{\Omega}}_{x}& =(x-\mu )[{F}_{1}({r}_{1})+{Q}_{1}(x,y,{r}_{1})]\\ & +(x-\mu +1)[{F}_{2}({r}_{2})+{Q}_{2}(x,y,{r}_{2})],\\ {\mathit{\Omega}}_{y}& =y[{G}_{1}({r}_{1})+{G}_{2}({r}_{2})+{Q}_{1}(x,y,{r}_{1})+{Q}_{2}(x,y,{r}_{2})],\end{array}$$(26)

where
$$\begin{array}{}{F}_{i}({r}_{i})={\mu}_{i}\left[{n}^{2}-\left({\displaystyle \frac{1}{{r}_{i}^{3}}}+{\displaystyle \frac{3{\delta}_{i}}{2{r}_{i}^{5}}}(2{\sigma}_{2i}+4{\sigma}_{1i}-{\sigma}_{3i})\right)\right],\\ {G}_{i}({r}_{i})={\mu}_{i}\left[{n}^{2}-\left({\displaystyle \frac{1}{{r}_{i}^{3}}}+{\displaystyle \frac{3{\delta}_{i}}{2{r}_{i}^{5}}}(4{\sigma}_{2i}+2{\sigma}_{1i}-{\sigma}_{3i})\right)\right],\\ {Q}_{i}(x,y,{r}_{i})={\displaystyle \frac{15{\mu}_{i}{\delta}_{i}}{2{r}_{i}^{7}}}\left[{\sigma}_{1i}{\left[x+(-1{)}^{i}{\mu}_{3-i}\right]}^{2}+{\sigma}_{2i}{y}^{2}\right],\end{array}$$(27)

and the mean motion will take the form
$${n}^{2}={\displaystyle 1+\frac{3}{2}\sum _{i=1}^{2}{\delta}_{i}(2{\sigma}_{2i}-{\sigma}_{1i}-{\sigma}_{3i}).}$$(28)

The location of the collinear points *L*_{i}, *i* = 1, 2, 3 is determined by *Ω*_{x} = *Ω*_{y} = 0 and *y* = 0. By Eqs. (26), (27) and (28) this property is translated in
$$\begin{array}{}f(x)=\left\{\begin{array}{l}x+{\displaystyle \frac{3x{\delta}_{1}}{2}}\left(2{\sigma}_{21}-{\sigma}_{11}-{\sigma}_{31}\right)\\ +{\displaystyle \frac{3x{\delta}_{2}}{2}}\left(2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32}\right)\\ -{\displaystyle \frac{(1-\mu )(x-\mu )}{{r}_{1}^{3}}}-{\displaystyle \frac{\mu (x-\mu +1)}{{r}_{2}^{3}}}\\ -{\displaystyle \frac{3{\delta}_{1}(1-\mu )(x-\mu )}{{r}_{1}^{5}}}(2{\sigma}_{21}-{\sigma}_{11}-{\sigma}_{31})\\ -{\displaystyle \frac{3{\delta}_{2}\mu (x-\mu +1)}{{r}_{2}^{5}}}(2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32})\end{array}\right\}=0,\end{array}$$(29)

where *r*_{1} = |*x* − *μ*| and *r*_{2} = |*x* − *μ* + 1|.

Hence we can rewrite Eq. (29) as
$$\begin{array}{}f(x)=\left\{\begin{array}{l}{f}_{1}(x)\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}-\mathrm{\infty}<x<\mu -\\ \\ {f}_{2}(x)\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}\mu -1<x<\mu \\ \\ {f}_{3}(x)\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}\mu <x<\mathrm{\infty}\end{array}\right.\end{array}$$(30)

where
$$\begin{array}{}{f}_{1}(x)=\left\{\begin{array}{l}x+{\displaystyle \frac{3x{\delta}_{1}}{2}}\left(2{\sigma}_{21}-{\sigma}_{11}-{\sigma}_{31}\right)\\ +{\displaystyle \frac{3x{\delta}_{2}}{2}}\left(2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32}\right)\\ \\ +{\displaystyle \frac{(1-\mu )}{(x-\mu {)}^{2}}}+{\displaystyle \frac{3{\delta}_{1}(1-\mu )}{(x-\mu {)}^{4}}}\left(2{\sigma}_{21}-{\sigma}_{11}-{\sigma}_{31}\right)\\ \\ +{\displaystyle \frac{\mu}{(x-\mu +1{)}^{2}}}+{\displaystyle \frac{3{\delta}_{2}\mu}{(x-\mu +1{)}^{4}}}\left(2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32}\right)\end{array}\right\}\end{array}$$(31)

$$\begin{array}{}{f}_{2}(x)=\left\{\begin{array}{l}x+{\displaystyle \frac{3x{\delta}_{1}}{2}}\left(2{\sigma}_{21}-{\sigma}_{11}-{\sigma}_{31}\right)\\ +{\displaystyle \frac{3x{\delta}_{2}}{2}}\left(2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32}\right)\\ \\ +{\displaystyle \frac{(1-\mu )}{(x-\mu {)}^{2}}}+{\displaystyle \frac{3{\delta}_{1}(1-\mu )}{(x-\mu {)}^{4}}}\left(2{\sigma}_{21}-{\sigma}_{11}-{\sigma}_{31}\right)\\ \\ -{\displaystyle \frac{\mu}{(x-\mu +1{)}^{2}}}-{\displaystyle \frac{3{\delta}_{2}\mu}{(x-\mu +1{)}^{4}}}\left(2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32}\right)\end{array}\right\}\end{array}$$(32)
$$\begin{array}{}{f}_{3}(x)=\left\{\begin{array}{l}x+{\displaystyle \frac{3x{\delta}_{1}}{2}}\left(2{\sigma}_{21}-{\sigma}_{21}-{\sigma}_{31}\right)\\ +{\displaystyle \frac{3x{\delta}_{2}}{2}}\left(2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32}\right)\\ \\ -{\displaystyle \frac{(1-\mu )}{(x-\mu {)}^{2}}}-{\displaystyle \frac{3{\delta}_{1}(1-\mu )}{(x-\mu {)}^{4}}}\left(2{\sigma}_{21}-{\sigma}_{11}-{\sigma}_{31}\right)\\ \\ -{\displaystyle \frac{\mu}{(x-\mu +1{)}^{2}}}-{\displaystyle \frac{3{\delta}_{2}\mu}{(x-\mu +1{)}^{4}}}\left(2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32}\right)\end{array}\right\}\end{array}$$(33)

To investigate the existence of libration collinear points and determine their locations we have to study the behavior of the function *f*. In this context the derivative of the function *f* will be controlled by
$$\begin{array}{}{f}^{\prime}(x)=\left\{\begin{array}{l}{f}_{1}^{\prime}(x)\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}-\mathrm{\infty}<x<\mu -1\\ \\ {f}_{2}^{\prime}(x)\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}\mu -1<x<\mu \\ \\ {f}_{3}^{\prime}(x)\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}\mu <x<\mathrm{\infty}\end{array}\right.\end{array}$$(34)

where the derivatives *f*′_{1}, *f*′_{2} and *f*′_{3} are given in the Appendix, see Eqs. (46–48).

Now Eqs. (46–48) show that the signs of the fifth and seventh terms of the function *f*′ will be effected by the values of (2σ_{2i} − σ_{1i} − σ_{3i}), (*i* = 1, 2), according to whether both of these values are positive or negative or with different signs, while the sum of the first three terms is not affected and will positive all the time because σ_{1i}, σ_{2i}, σ_{3i} ≪ 1 and *δ*_{i} < 1. Thereby we will investigate under which conditions *f*′(*x*) > 0 in the open intervals (−∞, *μ* − 1), (*μ* − 1, *μ*) and (*μ*, ∞) and we need to find the the necessary and sufficient conditions which makes
$$\begin{array}{}{f}^{\prime}(x)=\left\{\begin{array}{l}{f}_{1}^{\prime}(x)>0\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}-\mathrm{\infty}<x<\mu -1\\ \\ {f}_{2}^{\prime}(x)>0\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}\mu -1<x<\mu \\ \\ {f}_{3}^{\prime}(x)>0\phantom{\rule{0.1cm}{0ex}},\phantom{\rule{0.2cm}{0ex}}\mu <x<\mathrm{\infty}\end{array}\right.\end{array}$$(35)

In this context we may analyze five cases, which lead to *f*'(*x*) > 0, these cases are:

(σ_{21} ≥ σ_{11} ≥ σ_{31}) and (σ_{22} ≥ σ_{12} ≥ σ_{32}),

(2σ_{21} − σ_{11} − σ_{31}) ≥ 0 and (2σ_{22} − σ_{12} − σ_{32}) ≥ 0,

(2σ_{21} − σ_{11} − σ_{31}) ≤ 0 and (2σ_{22} − σ_{12} − σ_{32}) ≤ 0,

(2σ_{21} − σ_{11} − σ_{31}) ≥ 0 and (2σ_{22} − σ_{12} − σ_{32}) ≤ 0,

(2σ_{21} − σ_{11} − σ_{31}) ≤ 0 and (2σ_{22} − σ_{12} − σ_{32}) ≥ 0.

Since the parameters *μ*, *δ*_{i}, σ_{1i}, σ_{2i}, σ_{3i} as well as the quantity 1 − *μ* are positive, with regard to the previous five cases the necessary and sufficient conditions can be translated to the unequal equations (36–40) respectively.

$${\sigma}_{2i}\ge {\sigma}_{1i}\ge {\sigma}_{3i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2$$(36)
$${\sigma}_{2i}\ge {\displaystyle \frac{1}{2}}({\sigma}_{1i}+{\sigma}_{3i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2$$(37)
$${\delta}_{i}|2{\sigma}_{2i}-{\sigma}_{1i}-{\sigma}_{3i}|\le {\displaystyle \frac{1}{6}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2$$(38)
$${\delta}_{2}|2{\sigma}_{22}-{\sigma}_{12}-{\sigma}_{32}|\le {\displaystyle \frac{1}{6}}$$(39)
$${\delta}_{1}|2{\sigma}_{21}-{\sigma}_{11}-{\sigma}_{31}|\le {\displaystyle \frac{1}{6}}$$(40)

where the derivatives *f*′_{1}, *f*′_{2} and *f*′_{3} are given in Appendix for every case.

Hence *f*′(*x*) > 0 in each of the open intervals (−∞, *μ* − 1), (*μ* − 1, *μ*) and (*μ*, ∞) and it follows that *f*(*x*) is strictly increasing in these intervals too. In addition Eq. (30) shows that

$\underset{x\to -\mathrm{\infty}}{\text{Lim}}f(x)=-\mathrm{\infty}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}f[(\mu -1{)}^{+}]=f({\mu}^{+})=-\mathrm{\infty},$

$\underset{x\to \mathrm{\infty}}{\text{Lim}}f(x)=\mathrm{\infty}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}f[(\mu -1{)}^{-}]=f({\mu}^{-})=\mathrm{\infty}.$

Thereby in every open interval the the

*f*(

*x*) changes its sign one time from (− to + ), and we get

*f*(

*μ* − 2) < 0,

*f*(0) > 0 and

*f*(

*μ* + 1) > 0.

Hence, we deduce that, *f*(*μ* − 2) < 0, *f*[(*μ* − 1)^{−}] > 0;*f*[(*μ* − 1)^{+}] < 0, *f*(0) > 0 and *f*(*μ*^{+}) < 0, *f*(*μ* + 1) > 0 and we concludes that there are only three zeros for *f*(*x*), when *y* = 0, which lie in the intervals (*μ* − 2, *μ* − 1), (*μ* − 1, 0) and (*μ*, *μ* + 1). Hence, there are three collinear libration points that lie in the intervals(*μ* − 2, *μ* − 1), (*μ* − 1, 0) and (*μ*, *μ* + 1), respectively. We will denote these by the symbols *L*_{1}, *L*_{2} and *L*_{3}

Now we can establish that there exists one and only one real value for *x* in each of the open intervals (−∞, *μ* − 1), (*μ* − 1, *μ*) and (*μ*, ∞) such that *f*(*x*) = 0 and the function *f*(*x*) is strictly increasing in these intervals, when one of the five conditions is achieved, otherwise we may have more than three collinear points.

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