On the derivation of the equations of motion and field from the law of conservation of energy

Abstract

Abstract. The equations of motion in analytical mechanics and the equations in the theory of electromagnetic fields are derived using the principle of least action applied to the Lagrangian function. In this article, Hamilton’s and Lagrange’s equations are obtained without using variational principles. The equations determining the generalized momenta are derived under the condition that the energy is a function of the generalized coordinates and velocities, and the conservation of the system’s energy under changes in any degrees of freedom is also proven. It should be noted that, based on these assumptions, Hamilton’s equations were obtained with a completely undetermined Hamiltonian. To find the relations describing the dependence of momenta on coordinates and velocities, it is shown that the Lagrangian function must be determined by considering the system’s energy as a known quantity. It should be emphasized that all obtained results are valid for a phase space in which the degrees of freedom are maximally independent of each other. The methods applied in analytical mechanics were successfully used to derive the field theory equations in electrodynamics. More precisely, the energy of a charged particle in an electromagnetic field was determined for suitable values of the vector potential of the field. As suitable values, the longitudinal electric field and vector potentials corresponding to the positions of the particle were considered.