  The Hamilton-Jacobi equation (HJE) is one of the most elegant approach to
Lagrangian systems such as geometrical optics and classical mechanics,
establishing the duality between trajectories and waves and paving the way
naturally for the quantum mechanics. Usually, this formalism is taught at the
end of a course on analytical mechanics through its technical aspects and its
relation to canonical transformations. I propose that the teaching of this
subject be centered on this duality along the lines proposed here, and the
canonical transformations be taught only after some familiarity with the HJE
has been gained by the students.
