## Appendix A  
\begin{tabular}{l l} \(f_{1}\) & \(=\frac{\mu_{0}N^{2}A\cos\varphi}{4}\bigg{[}\frac{\left(I_{0}-k_{1}x-k_{2}\dot{x }\right)^{2}}{\left(c_{0}-x\cos(\alpha)+y\sin(\alpha)\right)^{2}}-\frac{\left( I_{0}+k_{1}x+k_{2}\dot{x}\right)^{2}}{\left(  
\[\begin{array}{ll}f_{4}&=\frac{\mu_{0}N^{2}A\cos\varphi}{4}\left[\frac{\left(l_{0} -k_{1y}-k_{2y}\right)^{2}}{\left(c_{0}-xsin(\alpha)-y\cos(\alpha)\right)^{2}}- \frac{\left(l_{0}+k_{1y}+k_{2}\dot{y}\right)^{2}}{\left(c_{0}+xsin(\alpha)+y \cos(\alpha)\right)^{2}}\right]\\ &=\frac{4}{c_{0}^{3}}\mu_{0}N^{2}A\cos(\varphi)\left[\left(-l_{0}^{2}c_{0}^{2} \sin(\alpha)\right)x+\left(k_{1}l_{0}c_{0}^{3}-l_{0}^{2}c_{0}^{2}\cos(\alpha )\right)y+\left(-2l_{0}^{2}\cos^{3}(\alpha)-k_{1}^{2}c_{0}^{2}\cos(\alpha) \right.\right.\\ &\left.\left.+3k_{1}l_{0}c_{0}\cos^{2}(\alpha)\right)y^{3}+\left(-2l_{0}^{2} \sin(\alpha)+2l_{0}^{2}\sin(\alpha)\cos^{2}(\alpha)\right)x^{3}+\left(3k_{1} l_{0}c_{0}-3k_{1}l_{0}c_{0}\cos^{2}(\alpha)\right.\\ &\left.-6l_{0}^{2}\cos(\alpha)+6l_{0}^{2}\cos^{3}(\alpha)\right)yx^{2}+\left(- 6l_{0}^{2}\cos^{2}(\alpha)\sin(\alpha)+6k_{1}l_{0}c_{0}\cos(\alpha)\sin( \alpha)-k_{1}^{2}c_{0}^{2}\sin(\alpha)\right)y^{2}x\\ &\left.+\left(3k_{2}l_{0}c_{0}-3k_{2}l_{0}c_{0}\cos^{2}(\alpha)\right)x^{2} \dot{y}+\left(-k_{2}^{2}c_{0}^{2}\sin(\alpha)\right)x\dot{y}^{2}+\left(-k_{2} ^{2}c_{0}^{2}\cos(\alpha)\right)y\dot{y}^{2}+\left(k_{2}l_{0}c_{0}^{3}\right) \dot{y}\\ &\left.+\left(3k_{2}l_{0}c_{0}\cos^{2}(\alpha)-2k_{1}k_{3}c_{0}^{2}\cos(\alpha )\right)y^{2}\dot{y}+\left(6k_{2}l_{0}c_{0}\cos(\alpha)\sin(\alpha)-2k_{1}k_{2} c_{0}^{2}\sin(\alpha)\right)y\dot{y}x\right]\\ &\left.f_{5}\right.&=\frac{\mu_{0}N^{2}A\cos\varphi}{4}\left[\frac{\left(l_{0} -k_{1y}-k_{2y}\right)^{2}}{\left(c_{0}-y\right)^{2}}-\frac{\left(l_{0}+k_{1y} +k_{2}\dot{y}\right)^{2}}{\left(c_{0}+y\right)^{2}}\right]=\frac{4}{c_{0}^{3} }\mu_{0}N^{2}A\cos(\varphi)\left[\left(-k_{1}l_{0}c_{0}^{3}-l_{0}^{2}c_{0}^{2} \right)y\right.\\ &\left.+\left(2l_{0}^{2}+k_{1}^{2}c_{0}^{2}-3k_{1}l_{0}c_{0}\right)y^{3}+ \left(k_{2}^{2}c_{0}^{2}\right)y\dot{y}^{2}+\left(-k_{2}l_{0}c_{0}^{3}\right) \dot{y}+\left(-3k_{2}l_{0}c_{0}+2k_{1}k_{3}c_{0}^{2}\right)y^{2}\dot{y}\right] \end{array}\]  
\[\begin{array}{ll}f_{6}&=\frac{\mu_{0}N^{2}A\cos\varphi}{4}\left[\frac{\left( l_{0}-k_{1y}-k_{2}\dot{y}\right)^{2}}{\left(c_{0}+xsin(\alpha)-y\cos(\alpha) \right)^{2}}-\frac{\left(l_{0}+k_{1y}+k_{2}\dot{y}\right)^{2}}{\left(c_{0}- xsin(\alpha)+y\cos(\alpha)\right)^{2}}\right]\\ &=\frac{4}{c_{0}^{3}}\mu_{0}N^{2}A\cos(\varphi)\left[\left(-l_{0}^{2}c_{0}^{2} \sin(\alpha)\right)x+\left(-k_{1}l_{0}c_{0}^{3}+l_{0}^{2}c_{0}^{2}\cos(\alpha )\right)y+\left(2l_{0}^{2}\cos^{3}(\alpha)+k_{1}^{2}c_{0}^{2}\cos(\alpha) \right.\right.\\ &\left.\left.-3k_{1}l_{0}c_{0}\cos^{2}(\alpha)\right)y^{3}+\left(-2l_{0}^{2} \sin(\alpha)+2l_{0}^{2}\sin(\alpha)\cos^{2}(\alpha)\right)x^{3}+\left(-3k_{1} l_{0}c_{0}+3k_{1}l_{0}c_{0}\cos^{2}(\alpha)\right.\\ &\left.\left.+6l_{0}^{2}\cos(\alpha)-6l_{0}^{2}\cos^{3}(\alpha)\right)y\dot{x}^{ 2}+\left(-6l_{0}^{2}\cos^{2}(\alpha)\sin(\alpha)+6k_{1}l_{0}c_{0}\cos(\alpha) \sin(\alpha)-k_{1}^{2}c_{0}^{2}\sin(\alpha)\right)y^{2}x\\ &\left.\left.+\left(-3k_{2}l_{0}c_{0}+3k_{2}l_{0}c_{0}\cos^{2}(\alpha)\right)x^{ 2}\dot{y}+\left(-k_{2}^{2}c_{0}^{2}\sin(\alpha)\right)x\dot{y}^{2}+\left(k_{2} ^{2}c_{0}^{2}\cos(\alpha)\right)y\dot{y}^{2}+\left(-k_{2}l_{0}c_{0}^{3}\right) \dot{y}\\ &\left.+\left(-3k_{2}l_{0}c_{0}\cos^{2}(\alpha)+2k_{1}k_{3}c_{0}^{2}\cos(\alpha )\right)y\dot{y}^{2}\dot{y}+\left(6k_{2}l_{0}c_{0}\cos(\alpha)\sin(\alpha)-2k_{1 }k_{2}c_{0}^{2}\sin(\alpha)\right)y\dot{y}x\right]\end{array}\]