Using Hamilton-Jacoby equation to study the relativistic theory of the electron and some of its applications
Abstract
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 relationship between the Hamilton and Lagrange functions( ), 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.
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