User:Physicsch/sandbox

$$V[r] = \frac{e^{-(\nu[r] +\lambda[r])}}{\epsilon[r] + p[r]} *\biggr[ (\epsilon[r] + p[r])( e^{\nu[r] +\lambda[r]})r W[r] \biggr]$$

the equations are:

$$\ddot{r}-r\dot{\theta}^{2}=C(U+Vcos({\omega}t))\sum_{n=0}^{\infty}n(4n+1)\biggr[\sum_{m=0}^{n} (-1)^{m}\frac{(4n-2m)!(1-(cos(\alpha))^{2n-2m+1})}{4^{n}m!(2n-m)!(2n-2m+1)!}\biggr]\frac{r^{2n-1}}{R^{2n}}P_{2n}(cos(\theta)) $$

$$r\ddot{\theta}+2\dot{r}\dot{\theta}=C(U+Vcos({\omega}t))\sum_{n=0}^{\infty}(4n+1)\biggr[\sum_{m=0}^{n} (-1)^{m}\frac{(4n-2m)!(1-(cos(\alpha))^{2n-2m+1})}{4^{n}m!(2n-m)!(2n-2m+1)!}\biggr]\frac{r^{2n-1}}{R^{2n}}{d \over d\theta}(P_{2n}(cos(\theta))) $$

which with this assignment: $$X_1=r$$

$$X_2=\theta$$

$$X_3=\dot{r}$$

$$X_4=\dot{\theta}$$

will become a simultaneous systems of ODEs:

$$\dot{X_1}=X_3$$

$$\dot{X_2}=X_4$$

$$\dot{X_3}=X_1X_4^{2}+C(U+Vcos({\omega}t))\sum_{n=0}^{\infty}n(4n+1)\biggr[\sum_{m=0}^{n} (-1)^{m}\frac{(4n-2m)!(1-(cos(\alpha))^{2n-2m+1})}{4^{n}m!(2n-m)!(2n-2m+1)!}\biggr]\frac{r^{2n-1}}{R^{2n}}P_{2n}(cos(\theta))$$

$$\dot{X_4}=\frac{-2X_3X_4}{X_1}+\frac{C(U+Vcos({\omega}t))\sum_{n=0}^{\infty}(4n+1)\biggr[\sum_{m=0}^{n} (-1)^{m} \frac{(4n-2m)!(1-(cos(\alpha))^{2n-2m+1})}{4^{n}m!(2n-m)!(2n-2m+1)!}\biggr]\frac{r^{2n-1}}{R^{2n}}{d \over d\theta}(P_{2n}(cos(\theta)))}{X_1}$$