In this article, we used the generalized Hamilton-Jacoby equation to study the
relative motion of the electron in the arbitrary electromagnetic field, depending on
the action function(the principle of the least action), taking into account the
rel
ationship between the Hamilton and Lagrange functions(H P v L ), starting
with the equations of energy and motion for electron in the theory of special
relativity, where the Lagrangian were chosen so that the principle of variation is
equal to zero, thus the Lagrange equation was verified. The first and second sets of
Hamilton's equations were obtained and then Hamilton's conservation law, ie
electron energy. Study of some applications of the Hamilton-Jacoby equation on the
free motion of the particle, circular motion and the adiabatic transformations was
discussed. Kepler problem of the hydrogen atom was then discussed in relativistic
theory. The equation of the motion path of the electron was calculated and the
energy of the vibration and the frequency of the vibration were calculated.
