Stability of a rhomboidal configuration with a central body

Abstract

We consider a planar system of five bodies respectively located at $\mathbf{r}_{0}$, $\mathbf{r}_{1}$, $\mathbf{r}_{2}$, $\mathbf{r}_{3}$ and $\mathbf{r}_{4}$ with masses $m_{0} \geq 0$, $m_{1}= m_{2}= m$ and $m_{3}= m_{4}= \tilde{m}$ moving on a plane. The body of mass $m_{0}$ is considered to be at the center of the configuration which is assumed to be the origin of the coordinate system. The other four bodies, called the primaries, form a rhombus configuration at all time. We assume that the mutual attraction among all the bodies is of the Newtonian type.

Appendix

1.1 7.1 Matrices of the linearization

$$\begin{array}{rl}A& =\left(\begin{array}{cccc}{(1+p_{{\theta }_1})}^2+U_{{\rho }_1{\rho }_1}^1& U_{{\rho }_2{\rho }_1}^1& U_{{\rho }_3{\rho }_1}^1& U_{{\rho }_4{\rho }_1}^1\\ U_{{\rho }_1{\rho }_2}^2& {(1+p_{{\theta }_2})}^2+U_{{\rho }_2{\rho }_2}^2& U_{{\rho }_3{\rho }_2}^2& U_{{\rho }_4{\rho }_2}^2\\ U_{{\rho }_1{\rho }_3}^3& U_{{\rho }_2{\rho }_3}^3& {(1+p_{{\theta }_3})}^2+U_{{\rho }_3{\rho }_3}^3& U_{{\rho }_4{\rho }_3}^3\\ U_{{\rho }_1{\rho }_4}^4& U_{{\rho }_2{\rho }_4}^4& U_{{\rho }_3{\rho }_4}^4& {(1+p_{{\theta }_4})}^2+U_{{\rho }_4{\rho }_4}^4\end{array}\right),\\ & =\left(\begin{array}{cccc}{(1+p_{{\theta }_1})}^2+a_{11}& a_{12}& a_{13}& a_{14}\\ a_{21}& {(1+p_{{\theta }_2})}^2+a_{22}& a_{23}& a_{24}\\ a_{31}& a_{32}& {(1+p_{{\theta }_3})}^2+a_{33}& a_{34}\\ a_{41}& a_{42}& a_{43}& {(1+p_{{\theta }_4})}^2+a_{44}\end{array}\right),\\ B& =\left(\begin{array}{cccc}{U}_{{\theta }_1{\rho }_1}^1& U_{{\theta }_2{\rho }_1}^1& U_{{\theta }_3{\rho }_1}^1& U_{{\theta }_4{\rho }_1}^1\\ U_{{\theta }_1{\rho }_2}^2& U_{{\theta }_2{\rho }_2}^2& U_{{\theta }_3{\rho }_2}^2& U_{{\theta }_4{\rho }_2}^2\\ U_{{\theta }_1{\rho }_3}^3& U_{{\theta }_2{\rho }_3}^3& U_{{\theta }_3{\rho }_3}^3& U_{{\theta }_4{\rho }_3}^3\\ U_{{\theta }_1{\rho }_4}^4& U_{{\theta }_2{\rho }_4}^4& U_{{\theta }_3{\rho }_4}^4& U_{{\theta }_4{\rho }_4}^4\end{array}\right)=\left(\begin{array}{cccc}{b}_{11}& b_{12}& b_{13}& b_{14}\\ b_{21}& b_{22}& b_{23}& b_{24}\\ b_{31}& b_{32}& b_{33}& b_{34}\\ b_{41}& b_{42}& b_{43}& b_{44}\end{array}\right),\\ C& =\left(\begin{array}{cccc}{U}_{z_1{\rho }_1}^1& U_{z_2{\rho }_1}^1& U_{z_3{\rho }_1}^1& U_{z_4{\rho }_1}^1\\ U_{z_1{\rho }_2}^2& U_{z_2{\rho }_2}^2& U_{z_3{\rho }_2}^2& U_{z_4{\rho }_2}^2\\ U_{z_1{\rho }_3}^3& U_{z_2{\rho }_3}^3& U_{z_3{\rho }_3}^3& U_{z_4{\rho }_3}^3\\ U_{z_1{\rho }_4}^4& U_{z_2{\rho }_4}^4& U_{z_3{\rho }_4}^4& U_{z_4{\rho }_4}^4\end{array}\right),\end{array}$$

$$\begin{array}{rl}D& =\left(\begin{array}{cccc}2{\rho }_1(1+p_{{\theta }_1})& 0& 0& 0\\ 0& 2{\rho }_2(1+p_{{\theta }_2})& 0& 0\\ 0& 0& 2{\rho }_3(1+p_{{\theta }_3})& 0\\ 0& 0& 0& 2{\rho }_4(1+p_{{\theta }_4})\end{array}\right),\\ E& =\left(\begin{array}{cccc}{\tilde{e}}_{11}& \frac{1}{{\rho }_1^2}{U}_{{\rho }_2{\theta }_1}^1& \frac{1}{{\rho }_1^2}{U}_{{\rho }_3{\theta }_1}^1& \frac{1}{{\rho }_1^2}{U}_{{\rho }_4{\theta }_1}^1\\ \frac{1}{{\rho }_2^2}{U}_{{\rho }_1{\theta }_2}^2& {\tilde{e}}_{22}& \frac{1}{{\rho }_2^2}{U}_{{\rho }_3{\theta }_2}^2& \frac{1}{{\rho }_2^2}{U}_{{\rho }_4{\theta }_2}^2\\ \frac{1}{{\rho }_3^2}{U}_{{\rho }_1{\theta }_3}^3& \frac{1}{{\rho }_3^2}{U}_{{\rho }_2{\theta }_3}^3& {\tilde{e}}_{33}& \frac{1}{{\rho }_3^2}{U}_{{\rho }_4{\theta }_3}^3\\ \frac{1}{{\rho }_4^2}{U}_{{\rho }_1{\theta }_4}^4& \frac{1}{{\rho }_4^2}{U}_{{\rho }_2{\theta }_4}^4& \frac{1}{{\rho }_4^2}{U}_{{\rho }_3{\theta }_4}^4& {\tilde{e}}_{44}\end{array}\right)=\left(\begin{array}{cccc}{\tilde{e}}_{11}& \frac{1}{{\rho }_1^2}{e}_{12}& \frac{1}{{\rho }_1^2}{e}_{13}& \frac{1}{{\rho }_1^2}{e}_{14}\\ \frac{1}{{\rho }_2^2}{e}_{21}& {\tilde{e}}_{22}& \frac{1}{{\rho }_2^2}{e}_{23}& \frac{1}{{\rho }_2^2}{e}_{24}\\ \frac{1}{{\rho }_3^2}{e}_{31}& \frac{1}{{\rho }_3^2}{e}_{32}& {\tilde{e}}_{33}& \frac{1}{{\rho }_3^2}{e}_{34}\\ \frac{1}{{\rho }_4^2}{e}_{41}& \frac{1}{{\rho }_4^2}{e}_{42}& \frac{1}{{\rho }_4^2}{e}_{43}& {\tilde{e}}_{44}\end{array}\right),\end{array}$$

where $\tilde{e}_{ii}=\frac{1}{\rho_{i}^{2}} (2p_{\rho_{i}}p_{ \theta_{i}}+2 p_{\theta_{i}}+U^{i}_{\rho_{i}\theta_{i}}-\frac{2U^{i} _{\theta_{i}}}{\rho_{i}} )$ for $i=1,\dots ,4$.

$$\begin{array}{rl}F& =\left(\begin{array}{cccc}\frac{1}{{\rho }_1^2}{U}_{{\theta }_1{\theta }_1}^1& \frac{1}{{\rho }_1^2}{U}_{{\theta }_2{\theta }_1}^1& \frac{1}{{\rho }_1^2}{U}_{{\theta }_3{\theta }_1}^1& \frac{1}{{\rho }_1^2}{U}_{{\theta }_4{\theta }_1}^1\\ \frac{1}{{\rho }_2^2}{U}_{{\theta }_1{\theta }_2}^2& \frac{1}{{\rho }_2^2}{U}_{{\theta }_2{\theta }_2}^2& \frac{1}{{\rho }_2^2}{U}_{{\theta }_3{\theta }_2}^2& \frac{1}{{\rho }_2^2}{U}_{{\theta }_4{\theta }_2}^2\\ \frac{1}{{\rho }_3^2}{U}_{{\theta }_1{\theta }_3}^3& \frac{1}{{\rho }_3^2}{U}_{{\theta }_2{\theta }_3}^3& \frac{1}{{\rho }_3^2}{U}_{{\theta }_3{\theta }_3}^3& \frac{1}{{\rho }_3^2}{U}_{{\theta }_4{\theta }_3}^3\\ \frac{1}{{\rho }_4^2}{U}_{{\theta }_1{\theta }_4}^4& \frac{1}{{\rho }_4^2}{U}_{{\theta }_2{\theta }_4}^4& \frac{1}{{\rho }_4^2}{U}_{{\theta }_3{\theta }_4}^4& \frac{1}{{\rho }_4^2}{U}_{{\theta }_4{\theta }_4}^4\end{array}\right)=\left(\begin{array}{cccc}\frac{1}{{\rho }_1^2}{f}_{11}& \frac{1}{{\rho }_1^2}{f}_{12}& \frac{1}{{\rho }_1^2}{f}_{13}& \frac{1}{{\rho }_1^2}{f}_{14}\\ \frac{1}{{\rho }_2^2}{f}_{21}& \frac{1}{{\rho }_2^2}{f}_{22}& \frac{1}{{\rho }_2^2}{f}_{23}& \frac{1}{{\rho }_2^2}{f}_{24}\\ \frac{1}{{\rho }_3^2}{f}_{31}& \frac{1}{{\rho }_3^2}{f}_{32}& \frac{1}{{\rho }_3^2}{f}_{33}& \frac{1}{{\rho }_3^2}{f}_{34}\\ \frac{1}{{\rho }_4^2}{f}_{41}& \frac{1}{{\rho }_4^2}{f}_{42}& \frac{1}{{\rho }_4^2}{f}_{43}& \frac{1}{{\rho }_4^2}{f}_{44}\end{array}\right),\\ G& =\left(\begin{array}{cccc}\frac{1}{{\rho }_1^2}{U}_{z_1{\theta }_1}^1& \frac{1}{{\rho }_1^2}{U}_{z_2{\theta }_1}^1& \frac{1}{{\rho }_1^2}{U}_{z_3{\theta }_1}^1& \frac{1}{{\rho }_1^2}{U}_{z_4{\theta }_1}^1\\ \frac{1}{{\rho }_2^2}{U}_{z_1{\theta }_2}^2& \frac{1}{{\rho }_2^2}{U}_{z_2{\theta }_2}^2& \frac{1}{{\rho }_2^2}{U}_{z_3{\theta }_2}^2& \frac{1}{{\rho }_2^2}{U}_{z_4{\theta }_2}^2\\ \frac{1}{{\rho }_3^2}{U}_{z_1{\theta }_3}^3& \frac{1}{{\rho }_3^2}{U}_{z_2{\theta }_3}^3& \frac{1}{{\rho }_3^2}{U}_{z_3{\theta }_3}^3& \frac{1}{{\rho }_3^2}{U}_{z_4{\theta }_3}^3\\ \frac{1}{{\rho }_4^2}{U}_{z_1{\theta }_4}^4& \frac{1}{{\rho }_4^2}{U}_{z_2{\theta }_4}^4& \frac{1}{{\rho }_4^2}{U}_{z_3{\theta }_4}^4& \frac{1}{{\rho }_4^2}{U}_{z_4{\theta }_4}^4\end{array}\right),\\ H& =\left(\begin{array}{cccc}-\frac{2}{{\rho }_1}(p_{{\theta }_1}+1)& 0& 0& 0\\ 0& -\frac{2}{{\rho }_2}(p_{{\theta }_2}+1)& 0& 0\\ 0& 0& -\frac{2}{\rho 3}(p_{{\theta }_3}+1)& 0\\ 0& 0& 0& -\frac{2}{{\rho }_4}(p_{{\theta }_4}+1)\end{array}\right),\\ J& =\left(\begin{array}{cccc}-\frac{2p_{{\rho }_1}}{{\rho }_1}& 0& 0& 0\\ 0& -\frac{2p_{{\rho }_2}}{{\rho }_2}& 0& 0\\ 0& 0& -\frac{2p_{{\rho }_3}}{{\rho }_3}& 0\\ 0& 0& 0& -\frac{2p_{{\rho }_4}}{{\rho }_4}\end{array}\right),K=\left(\begin{array}{cccc}{U}_{{\rho }_1{z}_1}^1& U_{{\rho }_2{z}_1}^1& U_{{\rho }_3{z}_1}^1& U_{{\rho }_4{z}_1}^1\\ U_{{\rho }_1{z}_2}^2& U_{{\rho }_2{z}_2}^2& U_{{\rho }_3{z}_2}^2& U_{{\rho }_4{z}_2}^2\\ U_{{\rho }_1{z}_3}^3& U_{{\rho }_2{z}_3}^3& U_{{\rho }_3{z}_3}^3& U_{{\rho }_4{z}_3}^3\\ U_{{\rho }_1{z}_4}^4& U_{{\rho }_2{z}_4}^4& U_{{\rho }_3{z}_4}^4& U_{{\rho }_4{z}_4}^4\end{array}\right),\\ L& =\left(\begin{array}{cccc}{U}_{{\theta }_1{z}_1}^1& U_{{\theta }_2{z}_1}^1& U_{{\theta }_3{z}_1}^1& U_{{\theta }_4{z}_1}^1\\ U_{{\theta }_1{z}_2}^2& U_{{\theta }_2{z}_2}^2& U_{{\theta }_3{z}_2}^2& U_{{\theta }_4{z}_2}^2\\ U_{{\theta }_1{z}_3}^3& U_{{\theta }_2{z}_3}^3& U_{{\theta }_3{z}_3}^3& U_{{\theta }_4{z}_3}^3\\ U_{{\theta }_1{z}_4}^4& U_{{\theta }_2{z}_4}^4& U_{{\theta }_3{z}_4}^4& U_{{\theta }_4{z}_4}^4\end{array}\right),\\ M& =\left(\begin{array}{cccc}{U}_{z_1{z}_1}^1& U_{z_2{z}_1}^1& U_{z_3{z}_1}^1& U_{z_4{z}_1}^1\\ U_{z_1{z}_2}^2& U_{z_2{z}_2}^2& U_{z_3{z}_2}^2& U_{z_4{z}_2}^2\\ U_{z_1{z}_3}^3& U_{z_2{z}_3}^3& U_{z_3{z}_3}^3& U_{z_4{z}_3}^3\\ U_{z_1{z}_4}^4& U_{z_2{z}_4}^4& U_{z_3{z}_4}^4& U_{z_4{z}_4}^4\end{array}\right)=\left(\begin{array}{cccc}{m}_{11}& m_{12}& m_{13}& m_{13}\\ m_{12}& m_{11}& m_{13}& m_{13}\\ m_{31}& m_{31}& m_{33}& m_{34}\\ m_{31}& m_{31}& m_{34}& m_{33}\end{array}\right).\end{array}$$

1.2 7.2 Partial derivatives of the potential $U_{j}$

The second partial derivatives of $U_{j}$ evaluated at a rhomboidal configuration are such that

$$\begin{aligned} &\!\begin{aligned}[b] \frac{\partial^{2}U_{1}}{\partial \rho^{2}_{1}}&= \frac{\partial^{2}U _{2}}{\partial \rho^{2}_{2}} \\ &= \frac{m}{4 \lambda^{3}} + \frac{2 (m + m _{0})}{\lambda^{3}} + \frac{6 \lambda^{2} \tilde{m}}{(\lambda^{2}+ 1)^{5/2}} - \frac{2 \tilde{m}}{(\lambda^{2} + 1)^{3/2}}, \end{aligned} \end{aligned}$$

(38)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial \rho_{1} \partial \rho_{2}}= \frac{ \partial^{2}U_{2}}{\partial \rho_{2} \partial \rho_{1}}= -\frac{7 m}{4 \lambda^{3}}, \end{aligned}$$

(39)

$$\begin{aligned} &\!\begin{aligned}[b] \frac{\partial^{2}U_{1}}{\partial \rho_{1} \partial \rho_{3}} &= \frac{ \partial^{2}U_{1}}{\partial \rho_{1} \partial \rho_{4}}= \frac{ \partial^{2}U_{2}}{\partial \rho_{2} \partial \rho_{3}}= \frac{ \partial^{2}U_{2}}{\partial \rho_{2} \partial \rho_{4}}= \frac{ \partial^{2}U_{3}}{\partial \rho_{3} \partial \rho_{1}} \\ &= \frac{\partial^{2}U_{3}}{\partial \rho_{3} \partial \rho_{2}}=\frac{ \partial^{2}U_{4}}{\partial \rho_{4} \partial \rho_{1}}= \frac{ \partial^{2}U_{4}}{\partial \rho_{4} \partial \rho_{2}} = \frac{3 \lambda \tilde{m}}{(\lambda^{2} + 1)^{5/2}}, \end{aligned} \end{aligned}$$

(40)

$$\begin{aligned} &\!\begin{aligned}[b] \frac{\partial^{2}U_{3}}{\partial \rho^{2}_{3}} &= \frac{\partial^{2}U _{4}}{\partial \rho^{2}_{4}} \\ &= \frac{6 m}{(\lambda^{2} + 1)^{5/2}} - \frac{2 m}{(\lambda^{2} + 1)^{3/2}} + \frac{\tilde{m}}{4} + 2(m_{0} + \tilde{m}), \end{aligned} \end{aligned}$$

(41)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial \rho_{3} \partial \rho_{4}}= \frac{ \partial^{2}U_{4}}{\partial \rho_{4} \partial \rho_{3}}= -\frac{7 \tilde{m}}{4}, \end{aligned}$$

(42)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial \theta_{1} \partial \rho_{1}}= \frac{ \partial^{2}U_{1}}{\partial \theta_{1} \partial \rho_{2}}= \frac{ \partial^{2}U_{2}}{\partial \theta_{2} \partial \rho_{1}}= \frac{ \partial^{2}U_{2}}{\partial \theta_{2} \partial \rho_{2}}=\frac{ \partial^{2}U_{3}}{\partial \theta_{3} \partial \rho_{3}}= 0, \end{aligned}$$

(43)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial \rho_{1} \partial \theta_{1}} =\frac{ \partial^{2}U_{1}}{\partial \rho_{1} \partial \theta_{2}}=0, \end{aligned}$$

(44)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial \rho_{1} \partial \theta_{3}}= -\frac{ \partial^{2}U_{1}}{\partial \rho_{1} \partial \theta_{4}}= \tilde{m} \biggl( 1+ \frac{2\lambda^{2}-1}{(\lambda^{2}+1)^{5/2}} \biggr) , \end{aligned}$$

(45)

$$\begin{aligned} &\!\begin{aligned}[b] \frac{\partial^{2}U_{1}}{\partial \theta_{1} \partial \rho_{3}} &= -\frac{ \partial^{2}U_{1}}{\partial \theta_{1} \partial \rho_{4}}= - \frac{ \partial^{2}U_{2}}{\partial \theta_{2} \partial \rho_{3}} \\ &= -\frac{\partial^{2}U_{3}}{\partial \theta_{3} \partial \rho_{1}}=-\frac{ \partial^{2}U_{3}}{\partial \theta_{3} \partial \rho_{2}}= -\frac{ \partial^{2}U_{3}}{\partial \theta_{4} \partial \rho_{1}} \\ &= \biggl( 2 - \frac{3 }{(\lambda^{2} + 1)^{5/2}} + \frac{1}{(\lambda ^{2} + 1)^{3/2}} \biggr) \lambda \tilde{m}, \end{aligned} \end{aligned}$$

(46)

$$\begin{aligned} & \frac{\partial^{2}U_{2}}{\partial \rho_{2} \partial \theta_{3}}= -\frac{ \partial^{2}U_{2}}{\partial \rho_{2} \partial \theta_{4}}= \biggl( -1+ \frac{2\lambda^{2}- 1}{(\lambda^{2} + 1)^{5/2}} \biggr) \tilde{m}, \end{aligned}$$

(47)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial \rho_{3} \partial \theta_{1}}= \biggl( \frac{-2}{\lambda^{3}} + \frac{(\lambda^{2} - 2 )}{(\lambda^{2} + 1)^{5/2}} \biggr) \lambda m, \end{aligned}$$

(48)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial \rho_{3} \partial \theta_{2}}= -\frac{ \lambda m (\lambda^{2} - 2 ) }{(\lambda^{2} + 1)^{5/2}}, \end{aligned}$$

(49)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial \rho_{3} \partial \theta_{3}}= \frac{2 m}{\lambda^{2}}, \end{aligned}$$

(50)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial \rho_{3} \partial \theta_{4}}= \frac{ \partial^{2}U_{4}}{\partial \rho_{4} \partial \theta_{3}}=\frac{ \partial^{2}U_{4}}{\partial \rho_{4} \partial \theta_{4}}= 0, \end{aligned}$$

(51)

$$\begin{aligned} & \frac{\partial^{2}U_{4}}{\partial \rho_{4} \partial \theta_{1}}= -\frac{ \partial^{2}U_{4}}{\partial \rho_{4} \partial \theta_{2}}= \biggl( \frac{1}{ \lambda^{3}} + \frac{(-\lambda^{2} + 2 )}{(\lambda^{2} + 1)^{5/2}} \biggr) \lambda m, \end{aligned}$$

(52)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial z^{2}_{1}}= \frac{\partial^{2}U _{2}}{\partial z^{2}_{2}}= - \frac{9 m + 8m_{0}}{8\lambda^{3}} - \frac{2 \tilde{m}}{(\lambda^{2} + 1)^{3/2}}, \end{aligned}$$

(53)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial z_{1} \partial z_{2}}= \frac{ \partial^{2}U_{2}}{\partial z_{2} \partial z_{1}}= -\frac{7 m}{8 \lambda^{3}}, \end{aligned}$$

(54)

$$\begin{aligned} &\!\begin{aligned}[b] \frac{\partial^{2}U_{1}}{\partial z_{1} \partial z_{3}} &= \frac{ \partial^{2}U_{1}}{\partial z_{1} \partial z_{4}}= \frac{\partial^{2}U _{2}}{\partial z_{2} \partial z_{3}} \\ &=\frac{\partial^{2}U_{2}}{\partial z_{2} \partial z_{4}}= \biggl( \frac{1}{( \lambda^{2} + 1)^{3/2}} - 1 \biggr) \tilde{m}, \end{aligned} \end{aligned}$$

(55)

$$\begin{aligned} &\!\begin{aligned}[b] \frac{\partial^{2}U_{3}}{\partial z_{3} \partial z_{1}} &= \frac{ \partial^{2}U_{3}}{\partial z_{3} \partial z_{2}} = \frac{\partial^{2}U _{4}}{\partial z_{4} \partial z_{1}} \\ & =\frac{\partial^{2}U_{4}}{\partial z_{4} \partial z_{2}}= \biggl( \frac{1}{( \lambda^{2} + 1)^{3/2}} - \frac{1}{\lambda^{3}} \biggr) m, \end{aligned} \end{aligned}$$

(56)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial z^{2}_{3}}= \frac{\partial^{2}U _{4}}{\partial z^{2}_{4}}= -\frac{2 m}{(\lambda^{2} + 1)^{3/2}} - \frac{9 \tilde{m}}{8} -m_{0}, \end{aligned}$$

(57)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial z_{3} \partial z_{4}}= \frac{ \partial^{2}U_{4}}{\partial z_{4} \partial z_{3}}= - \frac{7 \tilde{m}}{8}, \end{aligned}$$

(58)

$$\begin{aligned} & \frac{\partial^{2}U_{4}}{\partial \rho_{4} \partial \theta_{3}}= \frac{ \partial^{2}U_{4}}{\partial \rho_{4} \partial \theta_{4}}= 0, \end{aligned}$$

(59)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial \rho_{1} \partial z_{1}}= \frac{ \partial^{2}U_{1}}{\partial \rho_{1} \partial z_{2}}= \frac{\partial ^{2}U_{1}}{\partial \rho_{1} \partial z_{3}}=\frac{\partial^{2}U_{1}}{ \partial \rho_{1} \partial z_{4}}= 0, \end{aligned}$$

(60)

$$\begin{aligned} & \frac{\partial^{2}U_{2}}{\partial \rho_{2} \partial z_{1}}= \frac{ \partial^{2}U_{2}}{\partial \rho_{2} \partial z_{2}}= \frac{\partial ^{2}U_{2}}{\partial \rho_{2} \partial z_{3}}=\frac{\partial^{2}U_{2}}{ \partial \rho_{2} \partial z_{4}}= 0, \end{aligned}$$

(61)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial \rho_{3} \partial z_{1}}= \frac{ \partial^{2}U_{3}}{\partial \rho_{3} \partial z_{2}}= \frac{\partial ^{2}U_{3}}{\partial \rho_{3} \partial z_{3}}=\frac{\partial^{2}U_{3}}{ \partial \rho_{3} \partial z_{4}}= 0, \end{aligned}$$

(62)

$$\begin{aligned} & \frac{\partial^{2}U_{4}}{\partial \rho_{4} \partial z_{1}}= \frac{ \partial^{2}U_{4}}{\partial \rho_{4} \partial z_{2}}= \frac{\partial ^{2}U_{4}}{\partial \rho_{4} \partial z_{3}}= \frac{\partial^{2}U_{4}}{ \partial \rho_{4} \partial z_{4}}= 0, \end{aligned}$$

(63)

$$\begin{aligned} & \frac{\partial^{2}U_{2}}{\partial z_{2} \partial \theta_{1}}= \frac{ \partial^{2}U_{2}}{\partial z_{2} \partial \theta_{2}}= \frac{\partial ^{2}U_{2}}{\partial z_{2} \partial \theta_{3}}= \frac{\partial^{2}U _{2}}{\partial z_{2} \partial \theta_{4}}= 0, \end{aligned}$$

(64)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial z_{3} \partial \theta_{1}}= \frac{ \partial^{2}U_{3}}{\partial z_{3} \partial \theta_{2}}= \frac{\partial ^{2}U_{3}}{\partial z_{3} \partial \theta_{3}}= \frac{\partial^{2}U _{3}}{\partial z_{3} \partial \theta_{4}}= 0, \end{aligned}$$

(65)

$$\begin{aligned} & \frac{\partial^{2}U_{4}}{\partial z_{4} \partial \theta_{1}}= \frac{ \partial^{2}U_{4}}{\partial z_{4} \partial \theta_{2}}= \frac{\partial ^{2}U_{4}}{\partial z_{4} \partial \theta_{3}}= \frac{\partial^{2}U _{4}}{\partial z_{4} \partial \theta_{4}}= 0, \end{aligned}$$

(66)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial \theta^{2}_{1}}= \frac{\partial ^{2}U_{2}}{\partial \theta^{2}_{2}} = -\frac{7 m}{8\lambda }+ \frac{6 \lambda^{2} \tilde{m}}{(\lambda^{2}+1)^{5/2}}, \end{aligned}$$

(67)

$$\begin{aligned} & \frac{\partial^{2}U_{1}}{\partial \theta_{1} \partial \theta_{2}}= \frac{ \partial^{2}U_{2}}{\partial \theta_{2} \partial \theta_{1}}= \frac{7m}{8 \lambda }, \end{aligned}$$

(68)

$$\begin{aligned} &\!\begin{aligned}[b] \frac{\partial^{2}U_{1}}{\partial \theta_{1} \partial \theta_{3}} &= \frac{ \partial^{2}U_{1}}{\partial \theta_{1} \partial \theta_{4}}= \frac{ \partial^{2}U_{2}}{\partial \theta_{2} \partial \theta_{3}} \\ &= \frac{\partial^{2}U_{2}}{\partial \theta_{2} \partial \theta_{4}}= -\frac{3 \lambda^{2} \tilde{m}}{(\lambda^{2}+1)^{3/2}}, \end{aligned} \end{aligned}$$

(69)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial \theta^{2}_{3}}= \frac{\partial ^{2}U_{4}}{\partial \theta^{2}_{4}} = -\frac{7}{8} \tilde{m}+ \frac{6 \lambda^{2} m}{(\lambda^{2}+1)^{5/2}}, \end{aligned}$$

(70)

$$\begin{aligned} &\!\begin{aligned}[b] \frac{\partial^{2}U_{3}}{\partial \theta_{3} \partial \theta_{1}} &= \frac{ \partial^{2}U_{3}}{\partial \theta_{3} \partial \theta_{2}}= \frac{ \partial^{2}U_{4}}{\partial \theta_{4} \partial \theta_{1}} \\ &= \frac{\partial^{2}U_{4}}{\partial \theta_{4} \partial \theta_{2}}= -\frac{3 \lambda^{2} m}{(\lambda^{2}+1)^{3/2}}, \end{aligned} \end{aligned}$$

(71)

$$\begin{aligned} & \frac{\partial^{2}U_{3}}{\partial \theta_{3} \partial \theta_{4}}= \frac{ \partial^{2}U_{4}}{\partial \theta_{4} \partial \theta_{3}}= \frac{7 \tilde{m}}{8}. \end{aligned}$$

(72)