Description of magnetic field lines without arcana

This work is based on the variational principle for magnetic field lines introduced in 1983 by Cary and Littlejohn. The action principles for magnetic field lines and for Hamiltonian mechanics are recalled to be analogous. It is shown that the first one can be rigorously proved from first principles without analytical calculations. Not only the action principles are analogous, but also a change of canonical coordinates is recalled to be equivalent to a change of gauge. Furthermore, using the vector potential makes obvious the freedom in the choice of "time" for describing Hamiltonian dynamics. These features may be used for a new pedagogical and intuitive introduction to Hamiltonian mechanics. In the context of confined magnetic fields, the action principle for magnetic field lines makes practical calculations simpler and safer, with an intuitive background and allowing to keep a high degree of generality, as shown in the practical example of the calculation of the width of a magnetic island, analytically derived without any need of abstract Fourier components and independently of the choice of coordinates. Moreover, a new formula provides explicitly the Boozer and Hamada magnetic coordinates from action-angle coordinates.