I'm trying to work with the geodesics on an ellipsoid of revolution and determining whether they are closed or open. So far I have found the geodesics for the ellipsoid given this equation
The ellipse $\gamma(v) = (a\cos(v),b\sin(v))$ revolved around the z axis to get $r(u,v) = (a\cos(v)\cos(u),a\cos(v)\sin(u),b\sin(v))$
- The meridians ($u = constant$, $v = v(t)$)
The parallels ($v = constant$, $u = u(t)$) which also must satisfy $f'=0$ and $u' = constant$
This took some work but eventually the curves $u = c\int \frac{\sqrt{a^2\sin^2(v) +b^2\cos^2(v)}}{a\sin(v)\sqrt{a^2\sin^2(v)-c^2}}$dv where c is some constant.
Now I want to know how to determine if they are closed or not, I know that every compact surfaces has at least 1 closed geodesic and I also know that every surface topologically equivalent to the sphere has at least 3 closed geodesics I am just not sure which ones are which.
I know the equator is a geodesic that is closed but really I am stuck from there. I'm wondering if maybe all these geodesics are closed, I know that's an open problem but perhaps?
If you are trying to do it analytically. I would bet that it is impossible, unless geodesics is parallel to equator or is a meridian.
If you want to do it numerically, you can find a closed geodesic with wanted precision, however you cannot generally prove that some geodesic isn't closed with numeric methods, because you can calculate only its finite length. There is always a posibility that it will close itself on the next couple of loops.
So I would start with understanding better what do you want.