UDK 51-72;519.216.2;514.85 MECHANICAL SYSTEMS WITH RANDOM PERTURBATIONS ON NON-LINEAR CONFIGURATION SPACES S. V. Azarina* * , Yu. <...> E. Gliklikh** , A. V. Obukhovskiĭ*** Voronezh State Technical University ** Voronezh State University ***Voronezh Institute of Management, Marketing and Finances The mechanical systems given on non-linear configuration spaces - smooth manifolds - in terms of Newton’s second law and subjected to random perturbations of either forces or velocities, are considered. <...> The machinery of mean derivatives is applied for obtaining well-posed description of the systems and for their investigation. <...> KEY WORDS: mechanical systems; random perturbation of force; random perturbation of velocity; set-valued force; mean derivatives; differential inclusion; Langevin equation. 1. <...> It is a well-known fact that a second order differential equation expressing the Newton’s law in n xt t xt xt () ( () ())=, , , is represented a as a first order system on the space of doubled dimension П М У vt t x t vt () () =, , () ( () ()) = xt v t a . (1.1) We call the first equation of above system horizontal and the second one vertical. <...> Analogous split takes place in the general case of a mechanical system on non-linear configuration space (smooth manifold) M . <...> For such systems the Newton’s law is formulated in terms of covariant derivatives in the form D dt mt t mt mt =, , a where D dt (1.2) () ( () ()), is the covariant derivative of Levi—Civitб connection of Riemannian metric on M that determines the kinetic energy of system. <...> Here Newton’s law (1.2) is equivalent to equation © Азарина С. <...> Note that a random perturbation in Newton’s law can arise in the horizontal component, in the vertical component and in the both ones. <...> The vertical perturbation means the perturbation of force field while the horizontal one means the perturbation of velocity. <...> The notion of mean derivatives was introduced by Edward Nelson (see [15, 16, 17]) for the needs of stochastic mechanics (a version of quantum mechanics). <...> The equation of motion in this theory (called the Newton—Nelson equation) was the first ВЕСТНИК ВГУ, СЕРИЯ: ФИЗИКА. <...> E. Gliklikh, A. V. Obukhovskiĭ example of equations in mean derivatives. <...> In all above-mentioned <...>