This problem is pretty complicated, and I have fooled around with it in Maple but I don't understand the concepts well enough to work it out yet.
Suppose we have a simple hyperbolic orbit where there is a central mass M and a smaller mass m, and there is more than enough energy in this system for the smaller mass to whiz by and fly back out into space, reasonably deflected. For this problem we have at least two conservation principles to work with, one being total energy
[tex]E=\frac{1}{2}m\vec{v}^2-\frac{GMm}{r}+\frac{ml^2}{2r^2}[/tex]
[tex]\frac{d}{dt}E=0[/tex]
and total angular momentum
[tex]L=mr^2\dot\theta[/tex]
[tex]\frac{d}{dt}L=0[/tex]
I am attempting to solve for maximum [tex]\dot{r}[/tex], with as few variables remaining in the equation as possible.
I have attempted to solve for [tex]\ddot{r}=0[/tex] and simplify that, but it is really complex, due in part to the transformation to polar coordinates:
[tex]\vec{v}^2=\dot{r}^2+r\dot{\theta}^2[/tex]
If anyone knows of a solution or a way to work this out that they think I will be able to do, I would really appreciate the help.
|