On an Elementary Derivation of the Hamilton-Jacobi Equation from the Second Law of Newton

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It is shown that for a relativistic particle moving in an electromagnetic field its equations of motion written in a form of the second law of Newton can be reduced with the help of elementary operations to the Hamilton-Jacobi equation. The derivation is based on a possibility of transforming the equation of motion to a completely antisymmetric form. Moreover, by perturbing the Hamilton-Jacobi equation we obtain the principle of least action.The analogous procedure is easily extended to a general relativistic motion of a charged relativistic particle in an electromagnetic field. It sis also shown that the special-relativistic Hamilton-Jacobi equation for a free particle allows one to easily demonstrate the wave-particle duality inherent to this equation and, in addition, to obtain the operators of the four-momentum whose eigenvalues are the classical four-momentum