Study the Non-Linear Stability of Non-Collinear Libration Point in the Restricted Three-Body Configuration When the Shapes of the Primaries are Taken as Heterogeneous and Finite-Straight Segment

Abstract

The main focus of the present research work is to analyze the non-linear stability of the triangular equilibrium points ${{\mathcal{L}}_{4}}$ and ${{\mathcal{L}}_{5}}$ in the restricted three-body problem (R3BP). The condition of stability has been found out under the influence of the heterogeneous primary and a radiating finite-straight segment secondary and also under the effect by Coriolis as well as Centrifugal forces. This piece of research has been done by doing the normalization of the Hamiltonian in order to attained the Birkhoff’s normal form of the Hamiltonian, since normal forms of Hamiltonian are important to study the non-linear stability of equilibrium points. The conditions of KAM Theorem have been examined in the presence of resonance cases $\omega _{1}^{'} = 2\omega _{2}^{'}$ and $\omega _{1}^{'} = 3\omega _{2}^{'}$ and found that these conditions have been failed for three values of mass ratios ${{\mu }_{1}},$ ${{\mu }_{2}}$ and ${{\mu }_{3}}.$ Except these three values, ${{\mathcal{L}}_{4}}$ and ${{\mathcal{L}}_{5}}$ are stable in non-linear sense within the range of linear stability $0 < \mu < {{\mu }_{c}},$ where ${{\mu }_{c}}$ is the critical value of mass parameter $\mu .$ Consequently, in the presence of above mentioned purturbations the triangular equilibrium points are unstable for these three values of mass ratios.

Appendices

Appendix A

$$\begin{gathered} {{\tau }_{0}} = \frac{1}{{96}}\left( {12{{\gamma }^{2}} + 36{{\gamma }^{2}}{{J}_{1}} + 156{{J}_{1}} - 108{{J}_{2}}} \right. \\ \left. { + \,\,12{{\gamma }^{2}}S - 16\gamma S + 52S + 132} \right), \\ \end{gathered} $$

$${{\tau }_{1}} = - \frac{1}{{12\sqrt 3 }}\left( {15{{J}_{1}} - 21{{J}_{2}} - 4\gamma S + 5S + 18} \right),$$

$${{\tau }_{2}} = - \frac{1}{{12}}\left( { - 6\gamma + 15\gamma {{J}_{1}} - 12{{J}_{1}} + 45{{J}_{2}} + \gamma S + 4S} \right),$$

$${{\tau }_{3}} = - \frac{1}{{48}}\left( {36\gamma {{J}_{1}} + 63\gamma {{J}_{2}} + 12{{\gamma }^{2}}S + 12\gamma S} \right),$$

$$\begin{gathered} {{\tau }_{4}} = - \frac{1}{{48\sqrt 3 }} \\ \times \,\,\left( {216{{\gamma }^{2}}{{J}_{1}} - 108\gamma {{J}_{1}} + 405\gamma {{J}_{2}} + 36{{\gamma }^{2}}S + 36\gamma S} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{5}} = \frac{1}{{64}}\left( { - 168{{\gamma }^{2}}{{J}_{1}} + 84\gamma {{J}_{1}} + 120{{J}_{1}} - 315\gamma {{J}_{2}}} \right. \\ \left. { - \,\,114{{J}_{2}} - 28{{\gamma }^{2}}S - 28\gamma S + 40S + 24} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{6}} = \frac{1}{{32\sqrt 3 }}\left( {72\gamma + 204\gamma {{J}_{1}} + 144{{J}_{1}} + 231\gamma {{J}_{2}}} \right. \\ \left. { - \,\,450{{J}_{2}} + 44{{\gamma }^{2}}S + 116\gamma S - 48S} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{7}} = \frac{1}{{64}}\left( {264{{\gamma }^{2}}{{J}_{1}} - 132\gamma {{J}_{1}} + 264{{J}_{1}} + 495\gamma {{J}_{2}}} \right. \\ \left. { - \,\,78{{J}_{2}} + 44{{\gamma }^{2}}S + 12\gamma S + 88S + 72} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{8}} = \frac{1}{{384}}\left( {168\gamma + 1188\gamma {{J}_{1}} - 144{{J}_{1}} + 525\gamma {{J}_{2}}} \right. \\ \left. { + \,\,720{{J}_{2}} + 100{{\gamma }^{2}}S + 348\gamma S + 48S} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{9}} = \frac{1}{{128\sqrt 3 }}\left( {1800{{\gamma }^{2}}{{J}_{1}} - 900\gamma {{J}_{1}} - 1032{{J}_{1}} + 3375\gamma {{J}_{2}}} \right. \\ \left. { + \,\,2784{{J}_{2}} + 300{{\gamma }^{2}}S + 316\gamma S - 344S - 72} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{10}}} = - \frac{1}{{128}}\left( {264\gamma + 1524\gamma {{J}_{1}} + 528{{J}_{1}} + 945\gamma {{J}_{2}}} \right. \\ \left. { - \,\,840{{J}_{2}} + 180{{\gamma }^{2}}S + 684\gamma S - 176S} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{11}}} = \frac{1}{{128\sqrt 3 }}\left( {1080{{\gamma }^{2}}{{J}_{1}} - 540\gamma {{J}_{1}} + 552{{J}_{1}} + 2025\gamma {{J}_{2}}} \right. \\ \left. { - \,\,24{{J}_{2}} + 180{{\gamma }^{2}}S + 4\gamma S + 184S + 72} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{12}}} = \frac{1}{{3072}}\left( {2760{{\gamma }^{2}}{{J}_{1}} - 1380\gamma {{J}_{1}} - 6840{{J}_{1}}} \right. \\ + \,\,5175\gamma {{J}_{2}} + 15750{{J}_{2}} + 460{{\gamma }^{2}}S + 460\gamma S \\ \left. { - \,\,2280S - 888} \right){\text{,}} \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{13}}} = - \frac{5}{{768\sqrt 3 }}\left( {360\gamma + 4620\gamma {{J}_{1}} - 1296{{J}_{1}} + 903\gamma {{J}_{2}}} \right. \\ \left. { + \,\,7506{{J}_{2}} + 172{{\gamma }^{2}}S + 1108\gamma S + 432S} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{14}}} = - \frac{1}{{512}}\left( {5160{{\gamma }^{2}}{{J}_{1}} - 2580\gamma {{J}_{1}} - 9720{{J}_{1}}} \right. \\ + \,\,9675\gamma {{J}_{2}} + 22230{{J}_{2}} + 860{{\gamma }^{2}}S + 1340 \gamma S \\ \left. { - \,\,3240S - 984} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{15}}} = \frac{5}{{256\sqrt 3 }}\left( {216\gamma + 2100\gamma {{J}_{1}} + 432{{J}_{1}} + 777\gamma {{J}_{2}}} \right. \\ \left. { + \,\,774{{J}_{2}} + 148{{\gamma }^{2}}S + 844\gamma S - 144S} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{16}}} = \frac{1}{{1024}}\left( {4440{{\gamma }^{2}}{{J}_{1}} - 2220\gamma {{J}_{1}} + 1560{{J}_{1}}} \right. \\ + \,\,8325\gamma {{J}_{2}} + 2370{{J}_{2}} + 740{{\gamma }^{2}}S - 220\gamma S \\ + \,\,520S + 24. \\ \end{gathered} $$

$$\begin{gathered} {{t}_{1}} = 4\omega _{1}^{2} + 3,\,\,\,\,{{t}_{2}} = 4\omega _{2}^{2} + 3,\,\,\,\,{{l}_{1}} = \sqrt {4\omega _{1}^{2} + 9} , \\ {\text{and}}\,\,\,\,{{l}_{2}} = \sqrt {4\omega _{2}^{2} + 9} . \\ \end{gathered} $$

$$\begin{gathered} \omega _{{01}}^{'} = \frac{1}{{\sqrt 2 }}\left( {1 + \left( { - 6{{\mu }^{2}} + \frac{{9\mu }}{2} - \frac{3}{4}} \right){{J}_{1}}} \right. \\ \left. { + \,\,\left( {\frac{{45\mu }}{8} + \frac{3}{{16}}} \right){{J}_{2}} + \left( {\frac{\mu }{2} - {{\mu }^{2}}} \right)S} \right). \\ \end{gathered} $$

$$\begin{gathered} \omega _{{02}}^{'} = \frac{1}{{192\sqrt 2 }}\sqrt {(248\,832{{\mu }^{2}} - 248\,832\mu + 9216} \\ \overline { + \,\,(2\,018\,304{{\mu }^{2}} - 2\,363\,904\mu + 138\,240){{J}_{1}}} \\ \overline { + \,\,(1\,596\,672{{\mu }^{2}} - 611\,712\mu + 380\,160){{J}_{2}}} \\ \overline { + \,\,( - 608\,256{{\mu }^{3}} + 1\,815\,552{{\mu }^{2}}} \\ \overline { - \,\,1\,336\,320\mu + 110\,592)S)} . \\ \end{gathered} $$

$${{\tau }_{{130}}} = \frac{{{{l}_{1}}}}{{2k{{\omega }_{1}}}},$$

$$\begin{gathered} {{\tau }_{{131}}} = \frac{1}{{576{{k}^{5}}{{l}_{1}}\omega _{1}^{3}}}{{J}_{1}}\left( { - 9504\gamma \omega {{{\text{1}}}^{6}} + 216\gamma \omega _{1}^{4}} \right. \\ - \,\,22\,788\gamma \omega _{1}^{2} + 5103\gamma + 11{\kern 1pt} {\kern 1pt} 264\omega _{1}^{{10}} - 27{\kern 1pt} 264\omega _{1}^{8} \\ \left. { + \,\,10\,176\omega _{1}^{6} + 26\,344\omega _{1}^{4} - 5076\omega _{1}^{2} + 729} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{132}}} = \frac{{ - 1}}{{768{{k}^{5}}{{l}_{1}}\omega _{1}^{3}}}{{J}_{2}}\left( { - 47\,520\gamma \omega _{1}^{6} + 7560\gamma \omega _{1}^{4}} \right. \\ - \,\,84\,780\gamma \omega _{1}^{2} + 18\,225\gamma + 9856\omega _{1}^{8} + 41\,984\omega _{1}^{6} \\ \left. { - \,\,5832\omega _{1}^{4} + 155\,844\omega _{1}^{2} - 35\,721} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{133}}} = \frac{1}{{216{{k}^{5}}{{l}_{1}}\omega _{1}^{3}}}S\left( { - 528\gamma \omega _{1}^{8} - 1524\gamma \omega _{1}^{6}} \right. \\ + \,\,1404\gamma \omega _{1}^{4} - 3429\gamma \omega _{1}^{2} + 729\gamma + 704\omega _{1}^{{10}} \\ \left. { - \,\,2688\omega _{1}^{8} - 1620\omega _{1}^{6} + 5386\omega _{1}^{4} - 3861\omega _{1}^{2} + 729} \right), \\ \end{gathered} $$

$${{\tau }_{{210}}} = - \frac{{4{{\omega }_{1}}}}{{k{{l}_{1}}}},$$

$$\begin{gathered} {{\tau }_{{211}}} = \frac{1}{{72{{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}{{J}_{1}}\left( { - 9504\gamma \omega _{1}^{6} + 11\,016\gamma \omega _{1}^{4}} \right. \\ + \,\,1512\gamma \omega _{1}^{2} + 5103\gamma + 11\,264\omega _{1}^{{10}} - 15\,488\omega _{1}^{8} \\ \left. { + \,\,24\,896\omega _{1}^{6} - 4040\omega _{1}^{4} - 13\,824\omega _{1}^{2} + 729} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{212}}} = \frac{1}{{96{{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}{{J}_{2}}\left( {47\,520\gamma \omega _{1}^{6} - 48\,600\gamma \omega _{1}^{4}} \right. \\ - \,\,7560\gamma \omega _{1}^{2} - 18\,225\gamma + 9856\omega _{1}^{8} - 17\,344\omega _{1}^{6} \\ \left. { + \,\,34\,200\omega _{1}^{4} + 7776\omega _{1}^{2} + 35\,721} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{213}}} = \frac{1}{{27{{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}S\left( {528\gamma \omega _{1}^{8} - 204\gamma \omega _{1}^{6} + 540\gamma \omega _{1}^{4}} \right. \\ - \,\,27\gamma \omega _{1}^{2} + 729\gamma + 704\omega _{1}^{{10}} - 416\omega _{1}^{8} + 1220\omega _{1}^{6} \\ \left. { + \,\,1138\omega _{1}^{4} - 1917\omega _{1}^{2} + 729} \right), \\ \end{gathered} $$

$${{\tau }_{{230}}} = - \frac{{3\sqrt 3 \gamma }}{{2k{{l}_{1}}{{\omega }_{1}}}},$$

$$\begin{gathered} {{\tau }_{{231}}} = \frac{1}{{64\sqrt 3 {{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}{{J}_{1}}\left( {7168\gamma \omega _{1}^{{10}} - 3968\gamma \omega _{1}^{8}} \right. \\ + \,\,37\,632\gamma \omega _{1}^{6} - 66\,776\gamma \omega _{1}^{4} + 16\,092\gamma \omega _{1}^{2} - 729\gamma \\ - \,\,3584\omega _{1}^{{10}} + 9856\omega _{1}^{8} - 25056\omega _{1}^{6} + 33\,544\omega _{1}^{4} \\ \left. { + \,\,11\,340\omega _{1}^{2} - 5103} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{232}}} = \frac{{\sqrt 3 }}{{256{{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}{{J}_{2}}\left( {9856\gamma \omega _{1}^{8} - 8768\gamma \omega _{1}^{6}} \right. \\ + \,\,112\,712\gamma \omega _{1}^{4} + 109\,260\gamma \omega _{1}^{2} - 35\,721\gamma \\ + \,\,17\,920\omega _{1}^{{10}} - 45\,440\omega _{1}^{8} + 117\,600\omega _{1}^{6} \\ \left. { - \,\,150\,680\omega _{1}^{4} - 34\,020\omega _{1}^{2} + 18\,225} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{233}}} = \frac{1}{{216\sqrt 3 {{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}S\left( {4032\gamma \omega _{1}^{{10}} + 6624\gamma \omega _{1}^{8}} \right. \\ + \,\,14\,148\gamma \omega _{1}^{6} - 37\,386\gamma \omega _{1}^{4} + 30\,861\gamma \omega _{1}^{2} - 6561\gamma \\ + \,\,2816\omega _{1}^{{12}} - 4416\omega _{1}^{{10}} + 25616\omega _{1}^{8} + 13\,892\omega _{1}^{6} \\ \left. { - \,\,22\,140\omega _{1}^{4} + 30\,861\omega _{1}^{2} - 6561} \right), \\ \end{gathered} $$

$${{\tau }_{{310}}} = - \frac{{{{\omega }_{1}}\left( {4\omega _{1}^{2} + 1} \right)}}{{2k{{l}_{1}}}},$$

$$\begin{gathered} {{\tau }_{{311}}} = \frac{{ - 1}}{{576{{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}{{J}_{1}}\left( { - 24\,192\gamma \omega _{1}^{8} - 69\,120\gamma \omega _{1}^{6}} \right. \\ + \,\,182\,952\gamma \omega _{1}^{4} - 105\,516\gamma \omega _{1}^{2} - 5103\gamma + 28\,672\omega _{1}^{{12}} \\ + \,\,116\,224\omega _{1}^{{10}} - 143\,232\omega _{1}^{8} + 210\,656\omega _{1}^{6} \\ \left. { - \,\,246\,664\omega _{1}^{4} + 110\,052\omega _{1}^{2} - 729} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{312}}} = \frac{{ - 1}}{{768{{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}{{J}_{2}}\left( {120\,960\gamma \omega _{1}^{8} + 371\,520\gamma \omega _{1}^{6}} \right. \\ - \,\,830\,520\gamma \omega _{1}^{4} + 440\,100\gamma \omega _{1}^{2} + 18\,225\gamma \\ + \,\,39\,424\omega _{1}^{{10}} + 132\,480\omega _{1}^{8} - 37\,984\omega _{1}^{6} \\ \left. { + \,\,504\,792\omega _{1}^{4} - 470\,124\omega _{1}^{2} - 35\,721} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{313}}} = \frac{{ - 1}}{{216{{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}S\left( {2112\gamma \omega _{1}^{{10}} + 7776\gamma \omega _{1}^{8}} \right. \\ - \,\,2652\gamma \omega _{1}^{6} + 6264\gamma \omega _{1}^{4} - 7263\gamma \omega _{1}^{2} \\ - \,\,729\gamma + 1792\omega _{1}^{{12}} + 11\,200\omega _{1}^{{10}} - 2256\omega _{1}^{8} \\ \left. { - \,\,3676\omega _{1}^{6} - 526\omega _{1}^{4} + 945\omega _{1}^{2} - 729} \right), \\ \end{gathered} $$

$${{\tau }_{{330}}} = \frac{{3\sqrt 3 \gamma }}{{2k{{l}_{1}}{{\omega }_{1}}}},$$

$$\begin{gathered} {{\tau }_{{331}}} = \frac{{ - 1}}{{64\sqrt 3 {{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}{{J}_{1}}\left( {7168\gamma \omega _{1}^{{10}} - 10\,880\gamma \omega _{1}^{8}} \right. \\ + \,\,28\,992\gamma \omega _{1}^{6} - 52\,952\gamma \omega _{1}^{4} + 12\,204\gamma \omega _{1}^{2} \\ - \,\,729\gamma - 3584\omega _{1}^{{10}} + 9856\omega _{1}^{8} - 25\,056\omega _{1}^{6} \\ \left. { + \,\,33\,544\omega _{1}^{4} + 11\,340\omega _{1}^{2} - 5103} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{332}}} = \frac{{ - \sqrt 3 }}{{256{{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}{{J}_{2}}\left( {9856\gamma \omega _{1}^{8} - 8768\gamma \omega _{1}^{6}} \right. \\ + \,\,112\,712\gamma \omega _{1}^{4} + 109\,260\gamma \omega _{1}^{2} - 35\,721\gamma \\ + \,\,17\,920\omega _{1}^{{10}} - 45\,440\omega _{1}^{8} + 117\,600\omega _{1}^{6} \\ \left. { - \,\,150\,680\omega _{1}^{4} - 34\,020\omega _{1}^{2} + 18\,225} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{333}}} = \frac{{ - 1}}{{216\sqrt 3 {{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}S\left( {4032\gamma \omega _{1}^{{10}} - 1152\gamma \omega _{1}^{8}} \right. \\ + \,\,4428\gamma \omega _{1}^{6} - 21\,834\gamma \omega _{1}^{4} + 26\,487\gamma \omega _{1}^{2} \\ - \,\,6561\gamma + 2816\omega _{1}^{{12}} - 4416\omega _{1}^{{10}} + 25\,616\omega _{1}^{8} \\ \left. { + \,\,13\,892\omega _{1}^{6} - 22\,140\omega _{1}^{4} + 30\,861\omega _{1}^{2} - 6561} \right), \\ \end{gathered} $$

$${{\tau }_{{410}}} = \frac{{3\sqrt 3 \gamma {{\omega }_{1}}}}{{2k{{l}_{1}}}},$$

$$\begin{gathered} {{\tau }_{{411}}} = - \frac{1}{{64\sqrt 3 {{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}{{J}_{1}}\left( {11\,264\gamma \omega _{1}^{{10}} - 21\,632\gamma \omega _{1}^{8}} \right. \\ + \,\,17\,216\gamma \omega _{1}^{6} + 8248\gamma \omega _{1}^{4} - 17\,280\gamma \omega _{1}^{2} + 729\gamma \\ - \,\,5632\omega _{1}^{{10}} + 2944\omega _{1}^{8} - 26\,656\omega _{1}^{6} + 31\,576\omega _{1}^{4} \\ \left. { - \,\,6696\omega _{1}^{2} + 5103} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{412}}} = \frac{{ - \sqrt 3 }}{{256{{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}{{J}_{2}}\left( { - 9856\gamma \omega _{1}^{8} - 41\,984\gamma \omega _{1}^{6}} \right. \\ + \,\,73\,624\gamma \omega _{1}^{4} - 3312\gamma \omega _{1}^{2} + 35\,721\gamma + 28\,160\omega _{1}^{{10}} \\ - \,\,18\,560\omega _{1}^{8} + 119\,840\omega _{1}^{6} - 131\,720\omega _{1}^{4} \\ \left. { + \,\,24\,840\omega _{1}^{2} - 18\,225} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{413}}} = \frac{1}{{216\sqrt 3 {{k}^{5}}l_{1}^{3}{{\omega }_{1}}}}S\left( { - 6336\gamma \omega _{1}^{{10}} + 21\,024\gamma \omega _{1}^{8}} \right. \\ + \,\,10\,620\gamma \omega _{1}^{6} - 44\,802\gamma \omega _{1}^{4} + 26\,973\gamma \omega _{1}^{2} \\ - \,\,6561\gamma + 2816\omega _{1}^{{12}} + 5312\omega _{1}^{{10}} - 27\,888\omega _{1}^{8} \\ \left. { + \,\,18\,212\omega _{1}^{6} - 10\,332\omega _{1}^{4} + 3645\omega _{1}^{2} - 6561} \right), \\ \end{gathered} $$

$${{\tau }_{{430}}} = \frac{{9 - 4\omega _{1}^{2}}}{{2k{{l}_{1}}{{\omega }_{1}}}},$$

$$\begin{gathered} {{\tau }_{{431}}} = \frac{1}{{576{{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}{{J}_{1}}\left( { - 114\,048\gamma \omega _{1}^{8} + 3456\gamma \omega _{1}^{6}} \right. \\ - \,\,77\,112\gamma \omega _{1}^{4} - 143\,856\gamma \omega _{1}^{2} + 45\,927\gamma \\ + \,\,135\,168\omega _{1}^{{12}} - 103\,936\omega _{1}^{{10}} + 91\,264\omega _{1}^{8} \\ \left. { + \,\,187\,104\omega _{1}^{6} - 2664\omega _{1}^{4} - 1944\omega _{1}^{2} + 6561} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{432}}} = \frac{1}{{768{{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}{{J}_{2}}\left( {570\,240\gamma \omega _{1}^{8} + 8640\gamma \omega _{1}^{6}} \right. \\ + \,\,210\,600\gamma \omega _{1}^{4} + 544\,320\gamma \omega _{1}^{2} - 164\,025\gamma \\ + \,\,39\,424\omega _{1}^{{10}} - 395\,392\omega _{1}^{8} - 80\,928\omega _{1}^{6} \\ \left. { - \,\,508\,680\omega _{1}^{4} - 973\,944\omega _{1}^{2} + 321\,489} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\tau }_{{433}}} = \frac{1}{{216{{k}^{5}}l_{1}^{3}\omega _{1}^{3}}}S\left( {2112\gamma \omega _{1}^{{10}} - 12\,480\gamma \omega _{1}^{8}} \right. \\ - \,\,3780\gamma \omega _{1}^{6} - 1296\gamma \omega _{1}^{4} - 22113\gamma \omega _{1}^{2} + 6561\gamma \\ + \,\,8448\omega _{1}^{{12}} - 4288\omega _{1}^{{10}} - 8816\omega _{1}^{8} + 18876\omega _{1}^{6} \\ \left. { + \,\,4086\omega _{1}^{4} - 21\,627\omega _{1}^{2} + 6561} \right). \\ \end{gathered} $$

$$\begin{gathered} {{A}_{1}} = \frac{{21\gamma }}{{16}} + \left( {\frac{{297\gamma }}{{32}} - \frac{9}{8}} \right){{J}_{1}} + \left( {\frac{{525\gamma }}{{128}} + \frac{{45}}{8}} \right){{J}_{2}} \\ + \,\,\left( {\frac{{87\gamma }}{{32}} + \frac{{25\omega _{1}^{4}}}{{54}} - \frac{{25\omega _{1}^{2}}}{{54}} + \frac{{37}}{{32}}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} {{A}_{2}} = - \frac{{3\sqrt 3 }}{8} + \left( { - \frac{{75\sqrt 3 \gamma }}{{16}} + \frac{{50\omega _{1}^{4}}}{{3\sqrt 3 }} - \frac{{50\omega _{1}^{2}}}{{3\sqrt 3 }} + \frac{{55\sqrt 3 }}{{16}}} \right){{J}_{1}} \\ + \,\,\left( {\frac{{1125\sqrt 3 \gamma }}{{64}} + \frac{{29\sqrt 3 }}{2}} \right){{J}_{2}} \\ + \,\,\left( {\frac{{79\gamma }}{{16\sqrt 3 }} + \frac{{25\omega _{1}^{4}}}{{9\sqrt 3 }} - \frac{{25\omega _{1}^{2}}}{{9\sqrt 3 }} - \frac{5}{{4\sqrt 3 }}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} {{A}_{3}} = - \frac{{27\gamma }}{8} + \left( { - \frac{{\gamma \omega _{1}^{4}}}{6} + \frac{{\gamma \omega _{1}^{2}}}{6} - \frac{{2511\gamma }}{{64}} + \frac{{\omega _{1}^{4}}}{{12}}} \right. \\ \left. { - \,\,\frac{{\omega _{1}^{2}}}{{12}} + \frac{{117}}{{64}}} \right){{J}_{1}} + \left( {\frac{{1143\gamma }}{{128}} - \frac{{5\omega _{1}^{4}}}{{16}} + \frac{{5\omega _{1}^{2}}}{{16}} - \frac{{675}}{{64}}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{{\gamma \omega _{1}^{4}}}{{36}} + \frac{{\gamma \omega _{1}^{2}}}{{36}} - \frac{{399\gamma }}{{32}} - \frac{{7\omega _{1}^{4}}}{{18}} + \frac{{7\omega _{1}^{2}}}{{18}} - \frac{{39}}{{32}}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} {{B}_{1}} = - \frac{{33\gamma }}{{16}} + \left( { - \frac{{381\gamma }}{{32}} - \frac{{33}}{8}} \right){{J}_{1}} + \left( {\frac{{105}}{{16}} - \frac{{945\gamma }}{{128}}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{{171\gamma }}{{32}} - \frac{{5\omega _{1}^{4}}}{6} + \frac{{5\omega _{1}^{2}}}{6} - \frac{1}{{32}}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} {{B}_{2}} = - \frac{{9\sqrt 3 }}{8} + \left( {\frac{{135\sqrt 3 \gamma }}{{16}} - 10\sqrt 3 \omega _{1}^{4} + 10\sqrt 3 \omega _{1}^{2}} \right. \\ \left. { - \,\,\frac{{435\sqrt 3 }}{{16}}} \right){{J}_{1}} + \left( {\frac{{3\sqrt 3 }}{8} - \frac{{2025\sqrt 3 \gamma }}{{64}}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{{\sqrt 3 \gamma }}{{16}} - \frac{{5\omega _{1}^{4}}}{{\sqrt 3 }} + \frac{{5\omega _{1}^{2}}}{{\sqrt 3 }} - \frac{{25\sqrt 3 }}{4}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} {{B}_{3}} = \frac{{27\gamma }}{8} + \left( { - \frac{1}{6}7\gamma \omega _{1}^{4} + \frac{{7\gamma \omega _{1}^{2}}}{6} + \frac{{1827\gamma }}{{64}} + \frac{{7\omega _{1}^{4}}}{{12}} - \frac{{7\omega _{1}^{2}}}{{12}}} \right. \\ \left. { + \,\,\frac{{495}}{{64}}} \right){{J}_{1}} + \left( {\frac{{1017\gamma }}{{128}} - \frac{{35\omega _{1}^{4}}}{{16}} + \frac{{35\omega _{1}^{2}}}{{16}} - \frac{{675}}{{64}}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{{7\gamma \omega _{1}^{4}}}{{36}} + \frac{{7\gamma \omega _{1}^{2}}}{{36}} + \frac{{387\gamma }}{{32}} + \frac{{11\omega _{1}^{4}}}{6} - \frac{{11\omega _{1}^{2}}}{6} + \frac{{27}}{{32}}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} {{C}_{1}} = - \frac{{3\sqrt 3 }}{8} + \left( {\frac{{75\sqrt 3 {{\gamma }^{2}}}}{8} - \frac{{75\sqrt 3 \gamma }}{{16}} - \frac{{43\sqrt 3 }}{8}} \right){{J}_{1}} \\ + \,\,\left( {\frac{{1125\sqrt 3 \gamma }}{{64}} + \frac{{29\sqrt 3 }}{2}} \right){{J}_{2}} + \left( {\frac{{25\sqrt 3 {{\gamma }^{2}}}}{{16}} + \frac{{79\gamma }}{{16\sqrt 3 }} - \frac{{43}}{{8\sqrt 3 }}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} {{C}_{2}} = - \frac{{33\gamma }}{4} + \left( { - 60\gamma - \frac{{33}}{2}} \right){{J}_{1}} + \left( {\frac{{105}}{4} - \frac{{945\gamma }}{{32}}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{{45{{\gamma }^{2}}}}{8} - \frac{{51\gamma }}{2} + \frac{{11}}{2}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} {{C}_{3}} = \frac{{27\sqrt 3 }}{{32}} - \frac{{99\sqrt 3 }}{{32}}{{\gamma }^{2}} + \left( { - \frac{1}{8}357\sqrt 3 {{\gamma }^{2}} - \frac{{27\sqrt 3 \gamma }}{8}} \right. \\ \left. { + \,\,\frac{{243\sqrt 3 }}{{16}}} \right){{J}_{1}} + \left( { - 21\sqrt 3 {{\gamma }^{2}} - \frac{{585\sqrt 3 \gamma }}{{128}} - \frac{{4293\sqrt 3 }}{{128}}} \right){{J}_{2}} \\ + \,\,\left( { - 4\sqrt 3 {{\gamma }^{3}} - 16\sqrt 3 {{\gamma }^{2}} + \frac{{9\sqrt 3 \gamma }}{{16}} + \frac{{81\sqrt 3 }}{{16}}} \right)S. \\ \end{gathered} $$

$$\begin{gathered} A_{1}^{'} = - \frac{{3\sqrt 3 }}{{16}} + \left( {\frac{{75\sqrt 3 {{\gamma }^{2}}}}{{16}} - \frac{{75\sqrt 3 \gamma }}{{32}} - \frac{{43\sqrt 3 }}{{16}}} \right){{J}_{1}} \\ + \,\,\left( {\frac{{1125\sqrt 3 \gamma }}{{128}} + \frac{{29\sqrt 3 }}{4}} \right){{J}_{2}} \\ + \,\,\left( {\frac{{25\sqrt 3 {{\gamma }^{2}}}}{{32}} + \frac{{79\gamma }}{{32\sqrt 3 }} - \frac{{43}}{{16\sqrt 3 }}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} A_{2}^{'} = - \frac{{21\gamma }}{8} + \left( {\frac{9}{4} - \frac{{45\gamma }}{2}} \right){{J}_{1}} + \left( { - \frac{{525\gamma }}{{64}} - \frac{{45}}{4}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{{25{{\gamma }^{2}}}}{{16}} - \frac{{27\gamma }}{4} - \frac{3}{4}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} A_{3}^{'} = \frac{{9\sqrt 3 }}{{64}} + \frac{{63\sqrt 3 }}{{64}}{{\gamma }^{2}} + \left( {\frac{{21\sqrt 3 {{\gamma }^{2}}}}{4} + \frac{{27\sqrt 3 \gamma }}{8} + \frac{{87\sqrt 3 }}{{32}}} \right){{J}_{1}} \\ + \,\,\left( {\frac{{399\sqrt 3 {{\gamma }^{2}}}}{{64}} - \frac{{2655\sqrt 3 \gamma }}{{256}} - \frac{{1563\sqrt 3 }}{{256}}} \right){{J}_{2}} \\ + \,\,\left( {\frac{{19\sqrt 3 {{\gamma }^{3}}}}{{16}} + \frac{{23\sqrt 3 {{\gamma }^{2}}}}{8} - \frac{{37\sqrt 3 \gamma }}{{32}} + \frac{{29\sqrt 3 }}{{32}}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} B_{1}^{'} = \frac{{ - 9\sqrt 3 }}{{16}} + \left( { - \frac{1}{{16}}135\sqrt 3 {{\gamma }^{2}} + \frac{{135\sqrt 3 \gamma }}{{32}} - \frac{{69\sqrt 3 }}{{16}}} \right){{J}_{1}} \\ + \,\,\left( {\frac{{3\sqrt 3 }}{{16}} - \frac{{2025\sqrt 3 \gamma }}{{128}}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{1}{{32}}45\sqrt 3 {{\gamma }^{2}} - \frac{{\sqrt 3 \gamma }}{{32}} - \frac{{23\sqrt 3 }}{{16}}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} B_{2}^{'} = \frac{{33\gamma }}{8} + \left( {30\gamma + \frac{{33}}{4}} \right){{J}_{1}} + \left( {\frac{{945\gamma }}{{64}} - \frac{{105}}{8}} \right){{J}_{2}} \\ + \,\,\left( {\frac{{45{{\gamma }^{2}}}}{{16}} + \frac{{51\gamma }}{4} - \frac{{11}}{4}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} B_{3}^{'} = \frac{{27\sqrt 3 }}{{64}} - \frac{{99\sqrt 3 }}{{64}}{{\gamma }^{2}} + \left( { - \frac{1}{{16}}159\sqrt 3 {{\gamma }^{2}} - \frac{{63\sqrt 3 \gamma }}{8}} \right. \\ \left. { + \,\,\frac{{171\sqrt 3 }}{{32}}} \right){{J}_{1}} + \left( { - \frac{1}{2}21\sqrt 3 {{\gamma }^{2}} + \frac{{5355\sqrt 3 \gamma }}{{256}} - \frac{{549\sqrt 3 }}{{256}}} \right){{J}_{2}} \\ + \,\,\left( { - 2\sqrt 3 {{\gamma }^{3}} - \frac{{95\sqrt 3 {{\gamma }^{2}}}}{{16}} + \frac{{51\sqrt 3 \gamma }}{{32}} + \frac{{57\sqrt 3 }}{{32}}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} C_{1}^{'} = - \frac{{33\gamma }}{8} + \left( { - \frac{{381\gamma }}{{16}} - \frac{{33}}{4}} \right){{J}_{1}} + \left( {\frac{{105}}{8} - \frac{{945\gamma }}{{64}}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{{171\gamma }}{{16}} - \frac{{5\omega _{1}^{4}}}{3} + \frac{{5\omega _{1}^{2}}}{3} - \frac{1}{{16}}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} C_{2}^{'} = \frac{{3\sqrt 3 }}{4} + \left( {\frac{{75\sqrt 3 \gamma }}{8} - \frac{{100\omega _{1}^{4}}}{{3\sqrt 3 }} + \frac{{100\omega _{1}^{2}}}{{3\sqrt 3 }} - \frac{{55\sqrt 3 }}{8}} \right){{J}_{1}} \\ + \,\,\left( { - \frac{1}{{32}}1125\sqrt 3 \gamma - 29\sqrt 3 } \right){{J}_{2}} \\ + \,\,\left( { - \frac{{79\gamma }}{{8\sqrt 3 }} - \frac{{50\omega _{1}^{4}}}{{9\sqrt 3 }} + \frac{{50\omega _{1}^{2}}}{{9\sqrt 3 }} + \frac{5}{{2\sqrt 3 }}} \right)S, \\ \end{gathered} $$

$$\begin{gathered} C_{3}^{'} = \frac{{9\gamma }}{4} + \left( { - \frac{{\gamma \omega _{1}^{4}}}{3} + \frac{{\gamma \omega _{1}^{2}}}{3} + \frac{{585\gamma }}{{32}} + \frac{{\omega _{1}^{4}}}{6}} \right. \\ \left. { - \,\,\frac{{\omega _{1}^{2}}}{6} + \frac{{153}}{{32}}} \right){{J}_{1}} + \left( {\frac{{1683\gamma }}{{64}} - \frac{{5\omega _{1}^{4}}}{8} + \frac{{5\omega _{1}^{2}}}{8} - \frac{{45}}{8}} \right){{J}_{2}} \\ + \,\,\left( { - \frac{{\gamma \omega _{1}^{4}}}{{18}} + \frac{{\gamma \omega _{1}^{2}}}{{18}} + \frac{{123\gamma }}{{16}} + \omega _{1}^{4} - \omega _{1}^{2} + \frac{3}{{16}}} \right)S. \\ \end{gathered} $$

Appendix B

$${{\lambda }_{1}} = - \frac{{\left( {124\omega _{1}^{4} - 696\omega _{1}^{2} + 81} \right)\omega _{2}^{2}}}{{72{{k}^{4}}\left( {5\omega _{1}^{2} - 1} \right)}},$$

$$\begin{gathered} {{\lambda }_{2}} = \frac{1}{{5184{{k}^{8}}\omega _{1}^{2}{{{(5\omega _{1}^{2} - 1)}}^{2}}}}\left( { - 4\,035\,312\gamma \omega _{1}^{{12}}} \right. \\ + \,\,10\,137\,744\gamma \omega _{1}^{{10}} - 26\,489\,052\gamma \omega _{1}^{8} \\ + \,\,25\,840\,674\gamma \omega _{1}^{6} - 11\,490\,741\gamma \omega _{1}^{4} + 2\,081\,700\gamma \omega _{1}^{2} \\ - \,\,115\,911\gamma + 4\,782\,592\omega _{1}^{{16}} - 15\,177\,216\omega _{1}^{{14}} \\ + \,\,42\,383\,392\omega _{1}^{{12}} - 71\,013\,888\omega _{1}^{{10}} \\ + \,\,88\,451\,956\omega _{1}^{8} - 63\,758\,214\omega _{1}^{6} + 23\,719\,116\omega _{1}^{4} \\ \left. { - \,\,3\,975\,318\omega _{1}^{2} + 231\,822} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\lambda }_{3}} = - \frac{1}{{6912{{k}^{8}}l_{1}^{6}\omega _{1}^{2}{{{\left( {5\omega _{1}^{2} - 1} \right)}}^{2}}}}\left( {2\,424\,463\,360\gamma \omega _{1}^{{20}}} \right. \\ - \,\,5\,210\,711\,040\gamma \omega _{1}^{{18}} - 5\,281\,319\,680\gamma \omega _{1}^{{16}} \\ + \,\,116\,827\,200\gamma \omega _{1}^{{14}} - 16\,324\,148\,080\gamma \omega _{1}^{{12}} \\ - \,\,14\,186\,461\,200\gamma \omega _{1}^{{10}} - 12\,442\,142\,960\gamma \omega _{1}^{8} \\ + \,\,44\,394\,836\,220\gamma \omega _{1}^{6} - 31\,697\,352\,540\gamma \omega _{1}^{4} \\ + \,\,7\,373\,711\,070\gamma \omega _{1}^{2} - 272\,806\,380\gamma \\ - \,\,1\,146\,880\omega _{1}^{{20}} + 106\,358\,784\omega _{1}^{{18}} + 74\,665\,472\omega _{1}^{{16}} \\ - \,\,20\,873\,856\omega _{1}^{{14}} + 7\,454\,499\,104\omega _{1}^{{12}} \\ + \,\,18\,607\,904\,640\omega _{1}^{{10}} + 764\,789\,904\omega _{1}^{8} \\ - \,\,10\,925\,728\,578\omega _{1}^{6} + 11\,960\,040\,339\omega _{1}^{4} \\ \left. { - \,\,3\,079\,247\,886\omega _{1}^{2} + 167\,403\,915} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\lambda }_{4}} = \frac{1}{{15\,552{{k}^{8}}\omega _{1}^{2}{{{(5\omega _{1}^{2} - 1)}}^{2}}}}\left( {7680\gamma \omega _{1}^{{14}}} \right. \\ + \,\,6\,945\,840\gamma \omega _{1}^{{12}} - 12\,611\,760\gamma \omega _{1}^{{10}} + 25\,870\,680\gamma \omega _{1}^{8} \\ - \,\,26\,663\,172\gamma \omega _{1}^{6} + 10\,706\,391\gamma \omega _{1}^{4} - 1\,802\,898\gamma \omega _{1}^{2} \\ + \,\,115\,911\gamma + 2\,391\,296\omega _{1}^{{16}} - 6\,778\,368\omega _{1}^{{14}} \\ + \,\,16\,643\,312\omega _{1}^{{12}} - 29\,865\,936\omega _{1}^{{10}} + 39\,840\,584\omega _{1}^{8} \\ - \,\,29\,874\,612\omega _{1}^{6} + 10\,926\,087\omega _{1}^{4} - 1\,824\,930\omega _{1}^{2} \\ + \,\,115\,911), \\ \end{gathered} $$

$${{\lambda }_{5}} = - \frac{{{{\omega }_{1}}{{\omega }_{2}}\left( {64\omega _{1}^{2}\omega _{2}^{2} + 43} \right)}}{{6\left( {2\omega _{1}^{2} - 1} \right)\left( {5\omega _{1}^{2} - 1} \right)\left( {2\omega _{2}^{2} - 1} \right)\left( {5\omega _{2}^{2} - 1} \right)}},$$

$$\begin{gathered} {{\lambda }_{6}} = \frac{1}{{6912{{k}^{8}}l_{2}^{4}{{\omega }_{1}}\omega _{2}^{3}{{{(5{{\omega }_{1}}{{\omega }_{2}} - 2)}}^{2}}{{{(5{{\omega }_{1}}{{\omega }_{2}} + 2)}}^{2}}}} \\ \times \,\,\left( { - 4\,143\,052\,800\gamma \omega _{1}^{{22}} + 50\,548\,320\,000\gamma \omega _{1}^{{20}}} \right. \\ - \,\,292\,533\,113\,952\gamma \omega _{1}^{{18}} + 1\,044\,822\,333\,744\gamma \omega _{1}^{{16}} \\ - \,\,2\,533\,408\,170\,750\gamma \omega _{1}^{{14}} + 4\,099\,108\,158\,441\gamma \omega _{1}^{{12}} \\ - \,\,4\,344\,366\,562\,695\gamma \omega _{1}^{{10}} + 2\,975\,735\,955\,627\gamma \omega _{1}^{8} \\ - \,\,1\,275\,916\,988\,763\gamma \omega _{1}^{6} + 316\,536\,373\,800\gamma \omega _{1}^{4} \\ - \,\,38\,262\,811\,644\gamma \omega _{1}^{2} + 1\,641\,473\,424\gamma \\ + \,\,4\,910\,284\,800\omega _{1}^{{26}} - 64\,819\,404\,800\omega _{1}^{{24}} \\ + \,\,381\,133\,983\,232\omega _{1}^{{22}} - 1\,308\,895\,361\,536\omega _{1}^{{20}} \\ + \,\,3\,027\,950\,673\,472\omega _{1}^{{18}} - 5\,068\,970\,906\,000\omega _{1}^{{16}} \\ + \,\,6\,356\,325\,754\,182\omega _{1}^{{14}} - 6\,064\,313\,775\,697\omega _{1}^{{12}} \\ + \,\,4\,401\,863\,479\,895\omega _{1}^{{10}} - 2\,376\,831\,948\,731\omega _{1}^{8} \\ + \,\,901\,632\,336\,515\omega _{1}^{6} - 215\,606\,368\,144\omega _{1}^{4} \\ \left. { + \,\,26\,852\,210\,460\omega _{1}^{2} - 1\,264\,969\,872} \right), \\ {{\lambda }_{7}} = \frac{1}{{221\,184{{k}^{8}}l_{1}^{4}l_{2}^{4}\omega _{1}^{3}\omega _{2}^{3}{{{(5{{\omega }_{1}}{{\omega }_{2}} - 2)}}^{2}}{{{(5{{\omega }_{1}}{{\omega }_{2}} + 2)}}^{2}}}} \\ \times \,\,\left( {7\,954\,661\,376\,000\gamma \omega _{1}^{{28}} - 75\,665\,475\,993\,600\gamma \omega _{1}^{{26}}} \right. \\ + \,\,271\,796\,964\,618\,240\gamma \omega _{1}^{{24}} \\ - \,\,392\,452\,603\,600\,896\gamma \omega _{1}^{{22}} \\ - \,\,743\,019\,213\,445\,632\gamma \omega _{1}^{{20}} \\ + \,\,5\,026\,406\,733\,756\,288\gamma \omega _{1}^{{18}} \\ - \,\,8\,199\,977\,411\,126\,736\gamma \omega _{1}^{{16}} \\ + \,\,2\,480\,178\,180\,194\,208\gamma \omega _{1}^{{14}} \\ + \,\,7\,707\,731\,622\,963\,156\gamma \omega _{1}^{{12}} \\ - \,\,10\,860\,565\,766\,311\,674\gamma \omega _{1}^{{10}} \\ + \,\,6\,410\,650\,098\,483\,891\gamma \omega _{1}^{8} \\ - \,\,1\,817\,366\,957\,136\,288\gamma \omega _{1}^{6} \\ \end{gathered} $$

$$\begin{gathered} + \,\,190\,229\,194\,365\,615\gamma \omega _{1}^{4} + 1\,907\,613\,925\,524\gamma \omega _{1}^{2} \\ - \,\,819\,762\,622\,848\gamma + 3\,595\,147\,673\,600\omega _{1}^{{30}} \\ - \,\,36\,693\,604\,761\,600\omega _{1}^{{28}} + 177\,193\,300\,197\,376\omega _{1}^{{26}} \\ - \,\,408\,408\,655\,630\,336\omega _{1}^{{24}} + \,\,466\,874\,767\,163\,392\omega _{1}^{{22}} \\ - \,\,481\,277\,982\,848\,256\omega _{1}^{{20}} + \,\,722\,429\,204\,786\,944\omega _{1}^{{18}} \\ - \,\,469\,818\,883\,881\,072\omega _{1}^{{16}} - 2\,205\,579\,406\,937\,152\omega _{1}^{{14}} \\ + \,\,5\,331\,499\,990\,034\,668\omega _{1}^{{12}} - \,\,4710891159667894\omega _{1}^{{10}} \\ + \,\,2\,092\,219\,222\,798\,245\omega _{1}^{8} - 669\,951\,544\,657\,824\omega _{1}^{6} \\ + \,\,204\,250\,361\,728\,617\omega _{1}^{4} \\ \left. { - \,\,29\,806\,499\,049\,588\omega _{1}^{2} + 1\,138\,117\,039\,488} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\lambda }_{8}} = \frac{1}{{2592{{k}^{8}}l_{2}^{4}{{\omega }_{1}}\omega _{2}^{3}{{{(5{{\omega }_{1}}{{\omega }_{2}} - 2)}}^{2}}{{{(5{{\omega }_{1}}{{\omega }_{2}} + 2)}}^{2}}}} \times \\ \times \,\,\left( { - 1\,022\,856\,000\gamma \omega _{1}^{{22}} + 10\,573\,725\,600\gamma \omega _{1}^{{20}}} \right. \\ - \,\,44\,561\,688\,996\gamma \omega _{1}^{{18}} + 95\,725\,845\,264\gamma \omega _{1}^{{16}} \\ - \,\,110\,052\,799\,869\gamma \omega _{1}^{{14}} + 55\,003\,455\,603\gamma \omega _{1}^{{12}} \\ + \,\,14\,775\,636\,246\gamma \omega _{1}^{{10}} - 37\,352\,497\,374\gamma \omega _{1}^{8} \\ + \,\,24\,047\,980\,095\gamma \omega _{1}^{6} - 9\,101\,068\,221\gamma \omega _{1}^{4} \\ + \,\,{\text{2}}\,{\text{125}}\,{\text{150}}\,{\text{500}}\gamma \omega _{1}^{2} - 194\,895\,072\gamma \\ + \,\,{\text{306}}\,{\text{892}}\,{\text{800}}\omega _{1}^{{26}} - 4\,051\,212\,800\omega _{1}^{{24}} \\ + \,\,{\text{21}}\,{\text{710}}\,{\text{426}}\,{\text{752}}\omega _{1}^{{22}} - 58\,230\,013\,696\omega _{1}^{{20}} \\ + \,\,{\text{76}}\,{\text{873}}\,{\text{083}}\,{\text{496}}\omega _{1}^{{18}} - 11\,721\,065\,666\omega _{1}^{{16}} \\ - \,\,147\,577\,874\,007\omega _{1}^{{14}} + {\text{296}}\,{\text{579}}\,{\text{114}}\,{\text{015}}\omega _{1}^{{12}} \\ - \,\,307\,750\,140\,490\omega _{1}^{{10}} + {\text{192}}\,{\text{822}}\,{\text{509}}\,{\text{200}}\omega _{1}^{8} \\ - \,\,73\,067\,002\,675\omega _{1}^{6} + {\text{15}}\,{\text{572}}\,{\text{123}}\,{\text{675}}\omega _{1}^{4} \\ \left. { - \,\,1\,547\,915\,772\omega _{1}^{2} + {\text{47}}\,{\text{062}}\,{\text{944}}} \right), \\ \end{gathered} $$

$${{\lambda }_{9}} = - \frac{{\omega _{1}^{2}\left( {124\omega _{2}^{4} - 696\omega _{2}^{2} + 81} \right)}}{{72{{k}^{4}}\left( {5\omega _{2}^{2} - 1} \right)}},$$

$$\begin{gathered} {{\lambda }_{{10}}} = \frac{1}{{5184{{k}^{8}}\omega _{2}^{2}{{{\left( {5\omega _{2}^{2} - 1} \right)}}^{2}}}}\left( { - 4\,035\,312\gamma \omega _{2}^{{12}}} \right. \\ + \,\,{\text{10}}\,{\text{137}}\,{\text{744}}\gamma \omega _{2}^{{10}} - 26\,489\,052\gamma \omega _{2}^{8} \\ + {\text{25}}\,{\text{840}}\,{\text{674}}\gamma \omega _{2}^{6} - 11\,490\,741\gamma \omega _{2}^{4} \\ + \,\,{\text{2}}\,{\text{081}}\,{\text{700}}\gamma \omega _{2}^{2} - 115\,911\gamma + 4\,782\,592\omega _{2}^{{16}} \\ - \,\,15\,177\,216\omega _{2}^{{14}} + {\text{42}}\,{\text{383}}\,{\text{392}}\omega _{2}^{{12}} - 71\,013\,888\omega _{2}^{{10}} \\ + \,\,{\text{88}}\,{\text{451}}\,{\text{956}}\omega _{2}^{8} - 63\,758\,214\omega _{2}^{6} \\ \left. { + \,\,{\text{23}}\,{\text{719}}\,{\text{116}}\omega _{2}^{4} - 3\,975\,318\omega _{2}^{2} + 231\,822} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\lambda }_{{11}}} = \frac{1}{{6912{{k}^{8}}l_{2}^{6}\omega _{2}^{2}{{{\left( {5\omega _{2}^{2} - 1} \right)}}^{2}}}}\left( { - 2\,424\,463\,360\gamma \omega _{2}^{{20}}} \right. \\ + \,\,{\text{5}}\,{\text{210}}\,{\text{711}}\,{\text{040}}\gamma \omega _{2}^{{18}} + {\text{5}}\,{\text{281}}\,{\text{319}}\,{\text{680}}\gamma \omega _{2}^{{16}} \\ - \,\,116\,827\,200\gamma \omega _{2}^{{14}} + {\text{16}}\,{\text{324}}\,{\text{148}}\,{\text{080}}\gamma \omega _{2}^{{12}} \\ + \,\,{\text{14}}\,{\text{186}}\,{\text{461}}\,{\text{200}}\gamma \omega _{2}^{{10}} + {\text{12}}\,{\text{442}}\,{\text{142}}\,{\text{960}}\gamma \omega _{2}^{8} \\ - \,\,44\,394\,836\,220\gamma \omega _{2}^{6} + {\text{31}}\,{\text{697}}\,{\text{352}}\,{\text{540}}\gamma \omega _{2}^{4} \\ - \,\,7\,373\,711\,070\gamma \omega _{2}^{2} + {\text{272}}\,{\text{806}}\,{\text{380}}\gamma \\ + \,\,{\text{1}}\,{\text{146}}\,{\text{880}}\omega _{2}^{{20}} - 106\,358\,784\omega _{2}^{{18}} \\ - \,\,74\,665\,472\omega _{2}^{{16}} + {\text{20}}\,{\text{873}}\,{\text{856}}\omega _{2}^{{14}} \\ - \,\,7\,454\,499\,104\omega _{2}^{{12}} - 18\,607\,904\,640\omega _{2}^{{10}} \\ - \,\,764\,789\,904\omega _{2}^{8} + {\text{10}}\,{\text{925}}\,{\text{728}}\,{\text{578}}\omega _{2}^{6} \\ - \,\,11\,960\,040\,339\omega _{2}^{4} \\ \left. { + \,\,{\text{3}}\,{\text{079}}\,{\text{247}}\,{\text{886}}\omega _{2}^{2} - 167\,403\,915} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\lambda }_{{12}}} = \frac{1}{{15\,552{{k}^{8}}\omega _{2}^{2}{{{\left( {5\omega _{2}^{2} - 1} \right)}}^{2}}}}\left( {7680\gamma \omega _{2}^{{14}} + {\text{6}}\,{\text{945}}\,{\text{840}}\gamma \omega _{2}^{{12}}} \right. \\ - \,\,12\,611\,760\gamma \omega _{2}^{{10}} + {\text{25}}\,{\text{870}}\,{\text{680}}\gamma \omega _{2}^{8} - 26\,663\,172\gamma \omega _{2}^{6} \\ + \,\,{\text{10}}\,{\text{706}}\,{\text{391}}\gamma \omega _{2}^{4} - 1\,802\,898\gamma \omega _{2}^{2} + 115\,911\gamma \\ + \,\,{\text{2}}\,{\text{391}}\,{\text{296}}\omega _{2}^{{16}} - 6\,778\,368\omega _{2}^{{14}} + {\text{16}}\,{\text{643}}\,{\text{312}}\omega _{2}^{{12}} \\ - \,\,29\,865\,936\omega _{2}^{{10}} + {\text{39}}\,{\text{840}}\,{\text{584}}\omega _{2}^{8} - 29\,874\,612\omega _{2}^{6} \\ \left. { + \,\,{\text{10}}\,{\text{926}}\,{\text{087}}\omega _{2}^{4} - 1\,824\,930\omega _{2}^{2} + 115\,911} \right). \\ \end{gathered} $$

$$\begin{gathered} {{\nu }_{1}} = - \frac{1}{{3456{{k}^{8}}l_{2}^{4}\omega _{1}^{2}\omega _{2}^{2}{{{(5{{\omega }_{1}}{{\omega }_{2}} - 2)}}^{2}}{{{(5{{\omega }_{1}}{{\omega }_{2}} + 2)}}^{2}}}} \\ \times \,\,\left( { - 6\,366\,643\,200\gamma \omega _{1}^{{24}} + {\text{76}}\,{\text{119}}\,{\text{609}}\,{\text{600}}\gamma \omega _{1}^{{22}}} \right. \\ - \,\,423\,416\,494\,368\gamma \omega _{1}^{{20}} + {\text{1}}\,{\text{464}}\,{\text{413}}\,{\text{187}}\,{\text{312}}\gamma \omega _{1}^{{18}} \\ - \,\,3\,483\,591\,690\,366\gamma \omega _{1}^{{16}} + {\text{5}}\,{\text{610}}\,{\text{855}}\,{\text{551}}\,{\text{625}}\gamma \omega _{1}^{{14}} \\ - \,\,6\,004\,081\,529\,811\gamma \omega _{1}^{{12}} + {\text{4}}\,{\text{224}}\,{\text{080}}\,{\text{080}}\,{\text{287}}\gamma \omega _{1}^{{10}} \\ - \,\,1\,912\,601\,954\,637\gamma \omega _{1}^{8} + {\text{530}}\,{\text{631}}\,{\text{043}}\,{\text{200}}\gamma \omega _{1}^{6} \\ - \,\,82\,756\,473\,498\gamma \omega _{1}^{4} + {\text{6}}\,{\text{586}}\,{\text{030}}\,{\text{080}}\gamma \omega _{1}^{2} \\ - \,\,208\,948\,896\gamma + {\text{7}}\,{\text{545}}\,{\text{651}}\,{\text{200}}\omega _{1}^{{28}} \\ - \,\,97\,761\,484\,800\omega _{1}^{{26}}{\text{ + 565}}\,{\text{083}}\,{\text{033}}\,{\text{088}}\omega _{1}^{{24}} \\ - \,\,1\,944\,304\,135\,680\omega _{1}^{{22}} + {\text{4}}\,{\text{591}}\,{\text{455}}\,{\text{636}}\,{\text{416}}\omega _{1}^{{20}} \\ - \,\,7\,992\,409\,667\,600\omega _{1}^{{18}} + {\text{10}}\,{\text{662}}\,{\text{340}}\,{\text{941}}\,{\text{030}}\omega _{1}^{{16}} \\ - \,\,11\,057\,507\,932\,369\omega _{1}^{{14}} + {\text{8}}\,{\text{829}}\,{\text{328}}\,{\text{954}}\,{\text{935}}\omega _{1}^{{12}} \\ - \,\,5\,271\,937\,450\,459\omega _{1}^{{10}} + {\text{2}}\,{\text{247}}\,{\text{221}}\,{\text{681}}\,{\text{147}}\omega _{1}^{8} \\ - \,\,640\,211\,364\,704\omega _{1}^{6} + {\text{111}}\,{\text{489}}\,{\text{580}}\,{\text{820}}\omega _{1}^{4} \\ \left. { - \,\,10\,585\,058\,832\omega _{1}^{2} + {\text{417}}\,{\text{897}}\,{\text{792}}} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\nu }_{2}} = \frac{1}{{110\,592{{k}^{8}}l_{1}^{6}l_{2}^{6}\omega _{1}^{2}\omega _{2}^{2}{{{(5{{\omega }_{1}}{{\omega }_{2}} - 2)}}^{2}}{{{(5{{\omega }_{1}}{{\omega }_{2}} + 2)}}^{2}}}} \\ \times \,\,\left( {998\,299\,874\,099\,200\gamma \omega _{1}^{{32}}} \right. \\ - \,\,8\,306\,124\,534\,579\,200\gamma \omega _{1}^{{30}} \\ + \,\,{\text{26}}\,{\text{665}}\,{\text{318}}\,{\text{804}}\,{\text{684}}\,{\text{800}}\gamma \omega _{1}^{{28}} \\ - \,\,34\,087\,897\,459\,818\,496\gamma \omega _{1}^{{26}} \\ - \,\,25\,990\,461\,531\,058\,176\gamma \omega _{1}^{{24}} \\ + \,\,{\text{172}}\,{\text{727}}\,{\text{414}}\,{\text{711}}\,{\text{826}}\,{\text{432}}\gamma \omega _{1}^{{22}} \\ - \,\,182\,613\,749\,248\,943\,360\gamma \omega _{1}^{{20}} \\ - \,\,383\,573\,044\,575\,693\,952\gamma \omega _{1}^{{18}} \\ + \,\,{\text{94}}\,{\text{250}}\,{\text{075}}\,{\text{753}}\,{\text{205}}\,{\text{960}}\gamma \omega _{1}^{{16}} \\ - \,\,181\,904\,158\,349\,463\,296\gamma \omega _{1}^{{14}} \\ - \,\,1\,363\,275\,708\,631\,653\,492\gamma \omega _{1}^{{12}} \\ + \,\,{\text{1}}\,{\text{894}}\,{\text{491}}\,{\text{887}}\,{\text{408}}\,{\text{932}}\,{\text{914}}\gamma \omega _{1}^{{10}} \\ - \,\,1\,195\,290\,897\,411\,901\,919\gamma \omega _{1}^{8} \\ + \,\,{\text{398}}\,{\text{277}}\,{\text{829}}\,{\text{415}}\,{\text{828}}\,{\text{880}}\gamma \omega _{1}^{6} \\ - \,\,66\,407\,299\,541\,089\,947\gamma \omega _{1}^{4} \\ + \,\,{\text{4}}\,{\text{874}}\,{\text{339}}\,{\text{414}}\,{\text{802}}\,{\text{780}}\gamma \omega _{1}^{2} \\ - \,\,57\,522\,811\,042\,944\gamma \\ + \,\,57\,522\,362\,777\,600\omega _{1}^{{34}} \\ - \,\,663\,138\,625\,126\,400\omega _{1}^{{32}} \\ + \,\,{\text{3}}\,{\text{149}}\,{\text{706}}\,{\text{890}}\,{\text{838}}\,{\text{016}}\omega _{1}^{{30}} \\ - \,\,5\,363\,635\,079\,217\,152\omega _{1}^{{28}} \\ - \,\,7\,309\,594\,619\,674\,624\omega _{1}^{{26}} \\ {\text{ + }}\,\,{\text{36}}\,{\text{894}}\,{\text{469}}\,{\text{146}}\,{\text{263}}\,{\text{552}}\omega _{1}^{{24}} \\ - \,\,46\,738\,197\,622\,095\,872\omega _{1}^{{22}} \\ + \,\,{\text{37}}\,{\text{959}}\,{\text{089}}\,{\text{699}}\,{\text{583}}\,{\text{488}}\omega _{1}^{{20}} \\ - \,\,43\,207\,667\,883\,916\,288\omega _{1}^{{18}} \\ + \,\,{\text{57}}\,{\text{435}}\,{\text{679}}\,{\text{691}}\,{\text{611}}\,{\text{760}}\omega _{1}^{{16}} \\ + \,\,{\text{85}}\,{\text{820}}\,{\text{735}}\,{\text{703}}\,{\text{012}}\,{\text{640}}\omega _{1}^{{14}} \\ - \,\,277\,460\,386\,134\,250\,892\omega _{1}^{{12}} \\ + \,\,{\text{191}}\,{\text{575}}\,{\text{562}}\,{\text{528}}\,{\text{479}}\,{\text{230}}\omega _{1}^{{10}} \\ - \,\,17\,345\,806\,635\,014\,505\omega _{1}^{8} \\ - \,\,11\,126\,877\,658\,993\,008\omega _{1}^{6} \\ - \,\,3\,033\,351\,705\,639\,213\omega _{1}^{4} \\ + \,\,{\text{1}}\,{\text{036}}\,{\text{681}}\,{\text{878}}\,{\text{314}}\,{\text{628}}\omega _{1}^{2} \\ \left. { - \,\,39\,006\,374\,898\,816} \right), \\ \end{gathered} $$

$$\begin{gathered} {{\nu }_{3}} = - \frac{1}{{5184{{k}^{8}}l_{2}^{4}\omega _{1}^{2}\omega _{2}^{2}{{{(5{{\omega }_{1}}{{\omega }_{2}} - 2)}}^{2}}{{{(5{{\omega }_{1}}{{\omega }_{2}} + 2)}}^{2}}}} \\ \times \,\,\left( { - 2\,635\,411\,200\gamma \omega _{1}^{{24}} + {\text{25}}\,{\text{550}}\,{\text{755}}\,{\text{200}}\gamma \omega _{1}^{{22}}} \right. \\ - \,\,95\,913\,855\,408\gamma \omega _{1}^{{20}} + {\text{143}}\,{\text{537}}\,{\text{920}}\,{\text{608}}\gamma \omega _{1}^{{18}} \\ + \,\,{\text{35}}\,{\text{809}}\,{\text{658}}\,{\text{076}}\gamma \omega _{1}^{{16}} - 456\,652\,515\,684\gamma \omega _{1}^{{14}} \\ + \,\,{\text{750}}\,{\text{154}}\,{\text{966}}\,{\text{938}}\gamma \omega _{1}^{{12}} - 652\,561\,306\,230\gamma \omega _{1}^{{10}} \\ + \,\,{\text{351}}\,{\text{577}}\,{\text{961}}\,{\text{511}}\gamma \omega _{1}^{8} - 122\,748\,257\,256\gamma \omega _{1}^{6} \\ + \,\,{\text{26}}\,{\text{559}}\,{\text{441}}\,{\text{405}}\gamma \omega _{1}^{4} - 2\,869\,807\,752\gamma \omega _{1}^{2} \\ + \,\,{\text{104}}\,{\text{474}}\,{\text{448}}\gamma + {\text{1}}\,{\text{886}}\,{\text{412}}\,{\text{800}}\omega _{1}^{{28}} \\ - \,\,24\,440\,371\,200\omega _{1}^{{26}} + {\text{131}}\,{\text{347}}\,{\text{497}}\,{\text{472}}\omega _{1}^{{24}} \\ - \,\,374\,735\,320\,320\omega _{1}^{{22}} + {\text{617}}\,{\text{944}}\,{\text{051}}\,{\text{712}}\omega _{1}^{{20}} \\ - \,\,569\,347\,205\,480\omega _{1}^{{18}} + {\text{147}}\,{\text{911}}\,{\text{786}}\,{\text{388}}\omega _{1}^{{16}} \\ + \,\,{\text{307}}\,{\text{240}}\,{\text{212}}\,{\text{620}}\omega _{1}^{{14}} - 405\,652\,445\,934\omega _{1}^{{12}} \\ + \,\,{\text{199}}\,{\text{992}}\,{\text{203}}\,{\text{162}}\omega _{1}^{{10}} - 14\,434\,586\,137\omega _{1}^{8} \\ - \,\,28\,318\,307\,840\omega _{1}^{6} + {\text{12}}\,{\text{353}}\,{\text{226}}\,{\text{557}}\omega _{1}^{4} \\ \left. { - \,\,1\,937\,603\,592\omega _{1}^{2} + {\text{104}}\,{\text{474}}\,{\text{448}}} \right). \\ \end{gathered} $$