Say we have a spherical wire mesh raised to a negative voltage. Then let's say we release a proton from near the surface, and away from the surface, at some angle and speed. Also, imagine that the proton will always miss the mesh wires and pass through the surface.
The field in the center of this sphere is zero, so the proton travels in straight lines there, and outside of the sphere the proton travels in elliptical orbits with one focus located at the center of the sphere.
If the proton exits 3 times and winds up back at the same point it started, I imagine its trajectory would look something like this, where the dotted lines are the other part of the elliptical orbit that did not travel.
I wonder about a few things about this:
- Is the above idea possible in the first place?
- would it be possible to have a 2-exit orbit in the same sense?
- Is there only one combination of angle and velocity you can release it at to get a N-gon orbit?
- Alternatively, is there a relationship between angle and speed that fits such an N-gon orbit?
The above picture is imagining a "rounded" triangle orbit. If the last case is true (that there isn't a single unique N-gon orbit), then I wonder if there are varying degrees of roundness of the corners you could produce. If not, I imagine the orbital geometry would force all N-gon orbits to be similar to each other. I can't answer this easily to myself, and it seems such an interesting question.