## 2 Models of Motion  
This paper presents a technique for designing a low-energy trajectory for the Earth-Moon flight within the framework of a special planar auxiliary problem. We will consider the motion of the SC center of mass in the framework of the flat RC4 BP. Such simplified formulation of the well-known trajectory problem seems appropriate for two reasons: first--the solution obtained for it (numerically) can be used as an initial approximation of the main problem solution (without the considered assumptions). Second, when considering this version of the trajectory problem formulation, it is possible to find out some interesting properties and features from the obtained solution; which are also characterize the main problem. The latter, first of all, include the configuration of celestial bodies--the Earth, the Sun and the Moon, by using which it is possible to build the low-energy trajectory. In the general case, the motion of the SC center of mass (relative to the Earth along its flight trajectory to the Moon) can be considered within the framework of the restricted four-body problem (R4 BP). The latter, as is known, can be described by the following system of ordinary differential equations (undim., see Appendix A):  
\[\left. \begin{array}{l}\frac{d^{2}}{dt^{2}}\mathbf{r}=-\frac{\mu_{E}}{r^{3}} \mathbf{r}+\sum_{i=\{M,S\}}\mathbf{\Phi}_{i}\\ \mathbf{\Phi}_{i}=\mu_{i}\left[\frac{\mathbf{r}_{i}-\mathbf{r}}{|\mathbf{r}_{i }-\mathbf{r}_{i}|^{3}}-\frac{\mathbf{r}_{i}}{|\mathbf{r}_{i}|^{3}}\right] \end{array} \right\} \tag{1}\]  
where \(\mathbf{r}\) is the radius vector of the SC (vector function of time); \(\mathbf{r}_{i}\) is the radius vector characterizing the position of the \(i\)-th celestial body relative to the gravity center (Earth); \(\mathbf{\Phi}_{i}\)--perturbing acceleration from the \(i\)-th celestial body. Indexes \"\(E\)\", \"\(M\)\" and \"\(S\)\" corresponds to the Earth, Moon and Sun. It is important to note that system (1) describes the perturbed motion of the spacecraft's center of mass precisely in a relative form, not in an _absolute_ one. We will use the same relative form (describing the trajectory motion of the SC) in the course of all subsequent transformations while deriving the equations of motion of the SC. Further, according to the previously outlined approach to the problem of designing and analyzing low-energy SC trajectories in a simplified model, it is proposed to use next assumptions: trajectories of the SC and celestial bodies--the Moon, the Earth and the Sun--lies in the same plane. The orbits of the Sun and Moon in their relative motion (relative to the Earth) are circular. Taking into account the assumptions made, it turns out to be convenient to proceed to the description of the plane motion of the SC center of mass in the rotating (\"synodic\") coordinate system \(x\mathcal{C}y(z)\) (see Appendix B), synchronized with the motion of the Sun (see Figure 1). Then the main plane of the system--\(x\mathcal{C}y\)--is the ecliptic, the main direction (along the \(x\) axis) aims to the Sun. In accordance with this, the differential equations of system (1), which describes the motion of the SC, can be rewritten in the form:  
\[\left. \begin{array}{l}\frac{d^{2}}{dt^{2}}\mathbf{r}+2\left[n_{S}\mathbf{k}^ {0}\times\frac{\lambda}{dt}\mathbf{r}\right]-n_{S}{}^{2}\mathbf{r}=-\frac{n_{E }}{r^{3}}\mathbf{r}+\sum_{i=\{M,S\}}\mathbf{\Phi}_{i}\\ \mathbf{\Phi}_{i}=\mu_{i}\left[\frac{\mathbf{r}_{i}-\mathbf{r}}{|\mathbf{r}_{i }-\mathbf{r}_{i}|^{3}}-\frac{\mathbf{r}_{i}}{|\mathbf{r}_{i}|^{3}}\right] \end{array} \right\} \tag{2}\]or, in a scalar form, taking into account the lowering of the order of the system of differential equations:  
\[\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}}}+\[PERSON]_{M_{x}}+\[PERSON]_{S_{x}}\\ \frac{d}{dt}v_{y}+2v_{x}n_{S}-{n_{S}}^{2}y=-\frac{\mu_{E}y}{\sqrt{x^{2}+y^{2}}} +\Phi_{M_{y}}+\Phi_{S_{y}}\\ \frac{d}{dt}x=v_{x}\\ \frac{d}{dt}y=v_{y},\\ \mathbf{x}=\left(v_{x},v_{y},x,y\right)^{\mathbf{T}}\end{array}