User:Physicsch/sandbox

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

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:

${\dot {X_{1}}}=X_{3}$

${\dot {X_{2}}}=X_{4}$

${\dot {X_{3}}}=X_{1}X_{4}^{2}+C(U+Vcos({\omega }t))\sum _{n=0}^{\infty }n(4n+1){\biggr [}\sum _{m=0}^{n}(-1)^{m}{\frac {(4n-2m)!(1-(cos(\alpha ))^{2n-2m+1})}{4^{n}m!(2n-m)!(2n-2m+1)!}}{\biggr ]}{\frac {r^{2n-1}}{R^{2n}}}P_{2n}(cos(\theta ))$

${\dot {X_{4}}}={\frac {-2X_{3}X_{4}}{X_{1}}}+{\frac {C(U+Vcos({\omega }t))\sum _{n=0}^{\infty }(4n+1){\biggr [}\sum _{m=0}^{n}(-1)^{m}{\frac {(4n-2m)!(1-(cos(\alpha ))^{2n-2m+1})}{4^{n}m!(2n-m)!(2n-2m+1)!}}{\biggr ]}{\frac {r^{2n-1}}{R^{2n}}}{d \over d\theta }(P_{2n}(cos(\theta )))}{X_{1}}}$