Analogy between magnetic bottle and Van Allen's radiation belt A magnetic bottle is an arrangement that permits to confine charged particles. Here you can find a review for charged particle rotating in a magnetic field and at the bottom of the page a description of the magnetic bottle:
Motion of a charged particle in magnetic field
Let's use this image since it's more clear

What results from this picture is that a particle inside a magnetic bottle makes a circular motion following magnetic field lines which have a sort of cylindrical symmetry as it's expressed by this simulation:
Magnetic Bottle 1
So you can imagine to put the earth in the middle of the picture: in this way the two rings are to the two magnetic poles.
In the case of a Van Allen's radiation belt motion simulation (Radiation Belt as a Magnetic Bottle), particles move around a magnetic field lines instead of moving around the earth.
But from this simulation:
Magnetic Bottle 2 
you can see that a motion around a magnetic line is possible but i can't understand heuristically why.
So the questions are:


*

*how a particle can move around a magnetic line if there is closed to it another one? so how can a particle choose around which line move (losing in this way the cylindrical symmetry)?

*why the two type of motions are in disagreement?


Is the motion in the first picture just an approximation perhaps?
 A: In the Magnetic Bottle 1 YouTube video the charged particles are reflected from the ends of the bottle (where the ring are) because the magnetic field in non-uniform there. The particles tend to stay in the middle of the bottle where the field lines are roughly parallel.
The same thing happens in the Van Allen belts. You say "So you can imagine to put the earth in the middle of the picture" but this isn't a good comparison with the Earth's magnetic field. You have to imagine the bottle from the YouTube video being bent into a hoop with one end at the magnetic north pole and the other at the magnetic south pole.
I've tried to show this with these pictures:


You have to imagine bending the magnetic bottle into a U shape. I've tried to indicate this by showing where the middle and the ends of the bottle go when you bend it to match the Earths magnetic field. Just as the particles tend to stay in the middle of the bottle they tend to stay in the middle part of the Earths magnetic field. The particles are reflected from the non-uniform field near the north and south pole.
So the motion of the particles in the bottle is actually very similar to their motion in the Earth's field.
A: "how a particle can move around a magnetic line if there is closed to it another one? so how can a particle choose around which line move (losing in this way the cylindrical symmetry)?"
The simplest case is to look at a bundle of electrons emitting from a point (like water spreads out in a sprinkler head) in a uniform B field. So there is no particular field line that defines the axis of the electrons; it is the emitting point define the beginning of the axis and the field line meeting the emitting point the optical axis. Now you see these electrons spiral around the optical axis. If you only look at one electron, you lost the reference of center, therefore, which line to follow. 
