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In this article, we consider the problem of extending Hamilton’s principle to the class of natural mechanical systems with random perturbing forces of white noise type. By the method of moment functions, we construct the functionals taking a stationary value on the solutions of a given stochastic equation of Lagrangian structure.

The problem of constructing equivalent equations with a given structure of forces by the given system of stochastic equations is considered. The equivalence of equations in the sense of almost surely is investigated. The paper determines the conditions under which a given system of second-order Ito stochastic differential equations is represented in the form of stochastic Lagrange equations with non-potential forces of a certain structure. Necessary and sufficient conditions for the representability of stochastic equations in the form of stochastic equations with non-potential forces admitting the Rayleigh function are obtained. The obtained results are illustrated by an example of motion of a symmetric satellite in a circular orbit, assuming a change in pitch under the action of gravitational and aerodynamic forces.

Galiullin proposed a classification of inverse problems of dynamics for the class of ordinary differential equations (ODE). Considered problem belongs to the first type of inverse problems of dynamics (of the three main types of inverse problems of dynamics): the main inverse problem under the additional assumption of the presence of random perturbations. In this paper Hamilton and Birkhoff equations are constructed according to the given properties of motion in the presence of random perturbations from the class of processes with independent increments. The obtained necessary and sufficient conditions for the solvability of the problem of constructing stochastic differential equations of both Hamiltonian and Birkhoffian structure by the given properties of motion are illustrated by the example of the motion of an artificial Earth satellite under the action of gravitational and aerodynamic forces.

The austenitic–martensitic transformation in austenitic–ferritic duplex stainless steel CF8 subjected to cold plastic deformation with a deformation degree ε = 10–95% is studied here using transmission Mössbauer spectroscopy (MS), conversion electron Mössbauer spectroscopy (CEMS), and X-ray diffraction (XRD) methods. It is assumed that the α′-martensite phase appeared at ε > 10%. The CEMS results showed that the formation of α′-martensite occurred most intensively in the near-surface layers of the steel, distributing in depth with the growth of the deformation degree. The volume fraction of the α′-martensite was determined based on the results of calculations carried out via the MS and XRD methods, and a good correlation was observed. A modified Olson–Cohen model was proposed to determine the dependence of the amount of α′-martensite on the deformation degree ε. The coefficients included in the Olson–Cohen expression were found. [ABSTRACT FROM AUTHOR]

The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.

This research investigates the problems of self-identification of the Kazakh ethnic group in the post-totalitarian period, which are based on the values of traditional nomadic life of Kazakhs. The research shows the mechanisms of evolution of the ethnic group's worldview paradigms and their typology. The purpose of the research is to investigate the national image of Kazakhs' world through the structure of their artistic thinking, based on the assumption that art plays a significant role in the formation of an ethnic group's worldview. The research found that the significant factors in the structure of the ethnic worldview are traditional art, mythopoethic notions, language, and peculiarities of the national psychology and ethics. The features of mythopoethic reflection of reality in the artistic and spiritual practice of the nation, which are recorded in monuments of national art, were studied in detail. The research analyzed the typical features of the worldview of Ancient Turks as a prototype of the ethnic worldview of Kazakhs, which had a considerable effect on the structure of the Kazakh space-and-time continuum; the cults of ancestor, life and, death worshiping, which are relevant for the Kazakh mentality, were investigated. Special attention was paid to the main worldview cult of Ancient Turks-Tengri, which has numerous manifestation in the Kazakh culture: in cosmology, artistic thinking, ornamental, music, and epic art. The analysis allowed defining the ethnic worldview and concluding that the modern Kazakh culture is a synthesis of numerous elements of various cultures, which predetermines the exogenous nature of the traditional Kazakh worldview.

The inverse problem of reconstruction second-order Ito stochastic differential equations by the given properties of motion is solved when control is included in the drift coefficient. By using the separation method, the form of control parameters is determined that provides sufficient conditions for the existence of a given integral manifold for the nonlinear, linear, and scalar cases.

One of the inverse problems of dynamics in the presence of random perturbations is considered. This is the problem of the simultaneous construction of a set of first-order Ito stochastic differential equations with a given integral manifold, and a set of comparison functions. The given manifold is stable in probability with respect to these comparison functions.

We consider inverse problems for differential systems in the presence of random perturbations. By the method of additional variables, we establish sufficient conditions for the representation of stochastic differential equations of the second order in the form of stochastic Lagrange equations, as well as for the representation of stochastic differential equations of the first order in the form of stochastic canonical equations. The obtained results are illustrated by specific examples., Author(s): M. I. Tleubergenov [sup.1], D. T. Azhymbaev [sup.2] Author Affiliations: (1) grid.473156.2, Institute of Mathematics and Mathematical Modelling, , Pushkin Str., 125, 050010, Almaty, Kazakhstan (2) grid.443407.0, Zhubanov Aktyubinsk [...]

In this paper, a force field is constructed along a given integral manifold in the presence of random perturbing forces. In this case, two types of integral manifolds are considered separately: 1) trajectories that depend on generalized coordinates and do not depend on generalized velocities, and 2) trajectories that depend on both generalized coordinates and generalized velocities. The construction of the force field is carried out in the class of second-order stochastic Ito differential equations. It is assumed that the functions in the right-hand sides of the equation must be continuous in time and satisfy the Lipschitz condition in generalized coordinates and generalized velocities. Also this functions satisfy the condition for linear growth in generalized coordinates and generalized velocities.These assumptions ensure the existence and uniqueness up to stochastic equivalence of the solution to the Cauchy problem of the constructed equations in the phase space, which is a strictly Markov process continuous with probability 1. To solve the two posed problems, stochastic differential equations of perturbed motion with respect to the integral manifold are constructed. Moreover, in the case when the trajectories depend on generalized coordinates and do not depend on generalized velocities, the second order equations of perturbed motion are constructed, and in the case when the trajectories depend on both generalized coordinates and generalized velocities, the first order equations of perturbed motion are constructed. And further, in both cases by Erugin’s method necessary and sufficient conditions for solving the posed problems are derived.

Given second-order Ito stochastic equations, we construct almost surely equivalent stochastic equations of the Lagrangian structure. We establish the conditions for the direct and indirect analytical representations of the Lagrangian in the presence of random perturbations. The results obtained are illustrated by examples. [ABSTRACT FROM AUTHOR]

By the method of Lyapunov functions, we study the problem of optimal stabilization of an analytically defined integral manifold in the class of stochastic differential equations in the case where random perturbations belong to the class of processes with independent increments., UDC 517.925,519.216 One of the main requirements in the theory of inverse problems for the differential systems connected with the efficiency of a system is the requirement of stability of [...]

In the class of ordinary differential equations the following modification of the inverse problem of differential systems was previously considered: to construct both a set of systems of differential equations and a set of comparison functions for the given program motion. In this article, the modification of the inverse problem is considered in Stochastic case. In this problem it is assumed that random perturbations are from the class of processes with independent increments. By the given program of motion, two sets are constructed: the set of first - order Ito stochastic differential equations and the set of comparison functions. It is proved that there is a stability in probability of the given program motion with respect to the constructed comparison functions. To solve the problem Lyapunov functions method is used. Using Lyapunov’s second method makes it is possible to weaken the conditions imposed on the components of the constructed comparison functions, in contrast to the application of the Lyapunov characteristic numbers method for solving the inverse problem in the class of ordinary differential equations...

The Helmholtz problem is considered in a probabilistic formulation. By a given stochastic Langevin-Itoˆ equation in an indirect representation, as the equation of the Hamiltonian structure and the equation of the Birkhoffian structure are constructed. The functional that takes a stationary value on solutions of a given stochastic Birkhoff equation, is defined by the method of moment functions. The obtained results are illustrated by two examples: 1) the plane motion of a symmetric satellite in a circular orbit under the action of gravity and aerodynamic forces, and 2) the fluctuation motion of a gyroscope in a gimbal caused by the stochastic fluctuating moment of forces along the suspension axis of the inner ring.