Abstract
This article examines the application of the non-alternative variational principle to environmental mechanics under conditions of strong interactions of various physical fields. More precisely, this principle is considered as the main modeling method used to obtain the equations of motion of conservative and dissipative systems. Variations of functionals, conditions of stationarity of integral functionals, and variational principles in optics are presented. Calculations related to the principle of least action, Hamilton’s stationary action principle, Bernoulli’s variational equations, D’Alembert’s principle, and the variational equation of mechanics were carried out. It is shown that when transitioning from systems with a limited number of degrees of freedom to the environment, the kinetic and internal energies of the medium become functionals of the defining functions. It is proven that the first law of thermodynamics also represents a variational equation written through functionals in accordance with the variational principle. Variational approaches of analytical mechanics to discrete particles were applied to the study of oscillations of a thin string. By applying the stationarity condition for actions, the equation of string oscillation was obtained.