## 2 Models of Motion
\right\}\] (2a) where \[\mathbf{\Phi}_{M}=\mu_{M}\left( \begin{array}{l}\frac{x_{M}-x}{\sqrt{\left(x _{M}-x\right)^{2}+\left(y_{M}-y\right)^{2}}}-\frac{x_{M}}{\sqrt{x_{M}^{2}+y_{M }x^{2}}}\\ \frac{y_{M}-y}{\sqrt{\left(x_{M}-x\right)^{2}+\left(y_{M}-y\right)^{2}}}- \frac{y_{M}}{\sqrt{x_{M}^{2}+y_{M}x^{2}}}\end{array} \right),\mathbf{\Phi}_{S} =\mu_{S}\left( \begin{array}{l}\frac{r_{S}-x}{\sqrt{\left(r_{S}-x\right)^{2}+y ^{2}}}-\frac{1}{r_{S}^{2}}\\ \frac{-y}{\sqrt{\left(r_{S}-x\right)^{2}+y^{2}}}\end{array} \right)\] (3) vectors \(\mathbf{\Phi}_{M}\) and \(\mathbf{\Phi}_{S}\)--are perturbing accelerations of the SC (from the side of the Moon and the Sun). According to the accepted assumptions, in the above expressions (3) the value \(r_{S}\), which specifies the position of the Sun, located on the \(x\) axis of the uniformly rotating coordinate system \(xCy(z)\) is a constant. The absolute angular velocity of the axes rotation for the considered synodic coordinate system \(xCy(z)\) can be defined as (relative to the fixed direction of space) \[n_{S}\mathbf{k}^{0},\ n_{S}=\sqrt{\frac{\mu_{S}}{r_{S}{}^{3}}}\] where \(n_{S}\) is the mean motion of the Earth relative to the Sun (and the Sun relative to the Earth, respectively); \(\mathbf{k}^{0}\) is the unit vector of the coordinate system \(xCy(z)\), orthogonal to the main plane (i.e., the plane of motion). The model of the perturbed motion of the SC, described by relations (2), (2a), and (3) can be further simplified taking into account that the average distance between the SC center of mass and the Earth during its trajectory motion is at least two order smaller to the distance between the Earth and the Sun (\(r_{S}\)). This assumption is quite justified in view of the fact, as is known, a low-energy transfer trajectory to the Moon lies within the Earth's sphere of influence, and can only approach the boundary of it (however, like any other transfer trajectory to the Moon with a possible practical implementation). Then, assuming that  
\[\frac{\|x,y\|_{\mathbf{R}^{2}}}{r_{S}}\sim\varepsilon\]  
Figure 1: Coordinate systems: \(xCy(z)\)—geometric non-inertial, uniformly rotating, basic in terms of describing the motion of the center of mass of the spacecraft; \(\xi C\eta(z)\)—geocentric non-inertial, but with a constant direction of axes in inertial space (see Appendix B). The perturbing acceleration vector of the SC from the side of the Sun can be finally represented as follows  
\[\mathbf{\Phi}_{S}=\frac{\mu_{S}}{r_{S}{}^{3}}\bigg{(} \begin{array}{c}2x\\ -y\end{array} \bigg{)}+o(\varepsilon) \tag{4}\]  
Thus, relations (2) and (2a)-(4) completely describe the perturbed motion of the SC center of mass in a flat restricted circular four-body problem. Next, we will consider the perturbed motion of the SC center of mass in the accepted model on a basis of the unperturbed motion model corresponding to the restricted problem of two bodies: Earth and SC. The corresponding motion equations (according to the accepted model):  
\[\left. \begin{array}{l}\frac{d}{dt}v_{x}-2v_{y}n_{S}-n_{S}{}^{2}x=-\frac{\mu_ {E}\ x}{\sqrt{x^{2}+y^{2}}}\\ \frac{d}{dt}v_{y}+2v_{x}n_{S}-n_{S}{}^{2}y=-\frac{\mu_{E}y}{\sqrt{x^{2}+y^{2}} }\\ \frac{d}{dt}x=v_{x}\\ \frac{d}{dt}y=v_{y},\\ \mathbf{x}=\big{(}v_{x},v_{y},x,y\big{)}^{\mathbf{T}}\end{array} \right\} \tag{5}\]  
It is easy to show that the restricted circular two-body problem (RC2 BP) described by the system of differential Equation (5) naturally has a complete set of \"standard\" first mechanical integrals: energy \(h\), specific angular momentum \(\sigma\), and Laplace (\(\Lambda_{\xi}\), \(\Lambda_{\eta}\)). Let's write them down:  
\[\left.