# A Single-Launch Deployment Strategy for Lunar Constellations
## Appendix A Canonical Transformation for the Elliptic Restricted 4-Body Problem  
The Hamiltonian function characterizing the linear dynamics of the ER4 BP in the neighborhood of \(L_{1}\) is provided by Equation (14) and reported below  
\[ \begin{array}{c}H^{(\epsilon)}=\frac{1}{2}\big{(}p_{1}^{2}+p_{2}^{2}+p_{3} ^{2}\big{)}+p_{1}q_{2}-p_{2}q_{1}-\frac{\xi}{2}\big{(}2q_{1}^{2}-q_{2}^{2}-q_{ 3}^{2}\big{)}+\\ \qquad\qquad\qquad\qquad\qquad+e_{M}c_{1}\cos\theta_{E}+\frac{p_{0}}{r_{0}}c_{ 2}\end{array}  \tag{17}\]  
where the expressions for the coefficients \(c_{i}\) are  
\[ \begin{array}{c}c_{1}=-\cos\theta_{E}\Big{[}\frac{\mu_{1}}{r_{1}^{2}}+\frac {\mu_{2}}{r_{2}^{2}}+\frac{1}{2}\big{(}q_{1}^{2}+q_{2}^{2}+q_{3}^{2}\big{)} \Big{]}+\\ -\frac{\mu_{1}\mu_{2}a_{M}\cos\theta_{E}}{a_{E}}(q_{1}\cos\beta\cos\epsilon-q_ {2}\sin\beta\cos\epsilon+q_{3}\sin\epsilon)\bigg{(}\frac{1}{2|^{\kappa}}-\frac {1}{r_{1}^{2}|^{\kappa}}\bigg{)}+\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{\mu_{1}\mu_{2}a_{M}^{2}\cos \theta_{E}}{a_{E}^{2}}\bigg{(}\frac{\mu_{2}}{r_{2}^{2}|^{\kappa}}-\frac{\mu_ {1}}{r_{1}^{2}|^{\kappa}}\bigg{)}\end{array}  \tag{18}\]  
\[ \begin{array}{c}c_{2}=-\frac{\mu_{1}}{r_{1}^{3}|^{\kappa}}c_{3}-\frac{\mu_{ 2}}{r_{2}^{3}|^{\kappa}}c_{4}\end{array}  \tag{19}\]  
\[ \begin{array}{c}c_{3}=\frac{2a_{M}\mu_{2}\sin\epsilon(a_{E}q_{3}+a_{M}\mu_{2 }\sin\epsilon)}{a_{E}}-\frac{2(a_{E}q_{1}+a_{M}\mu_{2}\cos\beta\cos\epsilon)( a_{E}-a_{M}\mu_{2}\cos\beta\cos\epsilon)}{a_{E}}+\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{2a_{M}\mu_{2}\cos \epsilon\sin\beta(a_{E}q_{2}-a_{M}\mu_{2}\sin\beta\cos\epsilon)}{a_{E}}\end{array}  \tag{20}\]  
\[ \begin{array}{c}c_{4}=-\frac{2a_{M}\mu_{1}\sin\epsilon(a_{E}q_{3}-a_{M}\mu_{2 }\sin\epsilon)}{a_{E}}-\frac{2(a_{E}q_{1}+a_{M}\mu_{2}\cos\beta\cos\epsilon)( a_{E}+a_{M}\mu_{1}\cos\beta\cos\epsilon)}{a_{E}}+\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{2a_{M}\mu_{1}\cos\epsilon \sin\beta(a_{E}q_{2}+a_{M}\mu_{1}\sin\beta\cos\epsilon)}{a_{E}}\end{array}  \tag{21}\]  
\[ \begin{array}{c}\frac{\mu_{1}}{r_{1}^{\star}}=\frac{1-\mu}{|x_{L_{1}}+\mu|^{3 }}\bigg{(}q_{1}^{2}-\frac{1}{2}q_{2}^{2}-\frac{1}{2}q_{3}^{2}\bigg{)}\end{array}  \tag{22}\]  
\[ \begin{array}{c}\frac{\mu_{2}}{r_{2}^{\star}}=\frac{\mu}{|x_{L_{1}}+\mu-1|^{3 }}\bigg{(}q_{1}^{2}-\frac{1}{2}q_{2}^{2}-\frac{1}{2}q_{3}^{2}\bigg{)}\end{array}  \tag{23}\]\[\frac{\mu_{1}}{{r^{\ast}}_{1}^{3}}=\frac{1-\mu}{|x_{L_{1}}+\mu|^{5}}\bigg{(}q_{1} ^{2}-\frac{5}{2}q_{2}^{2}-\frac{5}{2}q_{3}^{2}\bigg{)} \tag{11}\]  
\[\frac{\mu_{2}}{{r^{\ast}}_{2}^{3}}=\frac{\mu}{|x_{L_{1}}+\mu-1|^{5}}\bigg{(}q_{1 }^{2}-\frac{5}{2}q_{2}^{2}-\frac{5}{2}q_{3}^{2}\bigg{)} \tag{12}\]  
\[\beta=\theta_{M}-\theta_{E} \tag{13}\]  
and \(\epsilon\) is the inclination of the Moon orbital plane with respect to the ecliptic plane. The canonical transformation rearranging \(H^{(\epsilon)}\) to a form equivalent to \(H\) can be developed from a type-3 generating function \(\mathcal{T}:[\mathbf{p}=\frac{\partial S}{\partial\mathbf{q}},\mathbf{Q}= \frac{\partial S}{\partial\mathbf{p}}]\)  
\[S(\mathbf{q},\mathbf{P})=q_{1}P_{1}+q_{2}P_{2}+q_{3}P_{3}+f_{1}q_{1}+f_{2}P_{ 2}+f_{3}q_{2}+f_{4}P_{1}+f_{5}q_{3}+f_{6}P_{3} \tag{14}\]  
The form for functions \(f_{i}\) is given below  
\[f_{i}(\theta_{M},\theta_{E})=b_{1}^{i}\cos\theta_{E}+b_{2}^{i}\sin\theta_{E}+ b_{3}^{i}\cos\beta+b_{4}^{i}\sin\beta\]  
and it depends on the unknown coefficients \(b_{j}^{i}\) determined as follows. First, the new coordinates are derived from \(S\)  
\[ \begin{cases}\mathbf{p}=\frac{\partial S}{\partial\mathbf{q}}\\ \mathbf{Q}=\frac{\partial S}{\partial\mathbf{p}}\end{cases} \rightarrow\]  
\[ \begin{cases}q_{1}=Q_{1}-f_{4}\\ q_{2}=Q_{2}-f_{2}\\ q_{3}=Q_{3}-f_{6}\end{cases}\begin{cases}p_{1}=P_{1}+f_{1}\\ p_{2}=P_{2}+f_{3}\\ p_{3}=P_{3}+f_{5}\end{cases}  \tag{15}\]  
Equation (15) is introduced into Equation (14) and rearranged as follows  
\[