Objective: the solution of problems of variational calculus is to find the differential equation that satisfies the condition, this applies to the fundamental lemma of variational calculus.Methods: The calculus of variations is one of the most important branches of applied mathematics, which played an important role in the development of particle mechanics. The calculus of variations - a branch of mathematics in which functions of the unknowns included in the integrand of some integral, selected a function in which the integral reaches its maximum or minimum value. Such tasks are called variational often occur at different stages of scientific and technological research.Despite the fact that Newtonian formulation commonly used in physics, in particular mechanics, there is an alternative way of describing the behavior of mechanical systems. This is Hamilton's principle, which consists in the fact that the true motion of a mechanical system is carried out in such a way that a certain integral, called the action takes a stationary value.Problems of mechanics can be equally well expressed in the form of variation, as well as with the help of the Newton equation of motion, however, is often a clearer understanding of the behavior of physical systems more easily achieved by using energy approach than using Newtonian formulations.Results. The article covers the basics of variational principles of mechanics applied to a material point is a brief historical background of variational methods mechanics, considered classical variational problem with fixed ends, shows the relationship of the law of conservation of energy to the principle of stationary action, as well as obtaining the characteristic function of a non-conservative system as an example of a mechanical system Linear friction. On the basis of Noether's theorem, shows the relationship of the first integral of the mechanical system, with its characteristic function, and interpretation made the first integral non-conservative mechanical system with linear friction.Keywords: Lagrange function, dynamical systems, applied mathematics, calculus of variations, Hamilton's principle.