[en] The paper considers the issues of kinematics and dynamics of a parallel mechanism and computer simulation of its movement. To solve the problems of dynamics of parallel mechanisms, the method of Lagrange equations of the second kind was used. In computer modeling, the Matlab and Simulink application were used. (paper)

[en] Published in summary form only

[en] The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel on the upper half space $${\int <!--\; ???\; -->}_{{\mathbb{R}}_{+}^n}{{\int <!--\; ???\; -->}_{\mathrm{\partial <!--\; ???\; -->}{\mathbb{R}}_{+}^n}f\left(\mathrm{\xi <!--\; ??\; -->}\right)P\left(x,\mathrm{\xi <!--\; ??\; -->},\mathrm{\alpha <!--\; ??\; -->}\right)g\left(x\right)d\mathrm{\xi <!--\; ??\; -->}dx\le <!--\; ???\; -->C_{n,\mathrm{\alpha <!--\; ??\; -->},p,q^{\prime }}\parallel <!--\; ???\; -->g\parallel <!--\; ???\; -->}_{L{q}^{\prime }\left({\mathbb{R}}_{+}^n\right)}\parallel <!--\; ???\; -->f{\parallel <!--\; ???\; -->}_{L{q}^{\prime }\left(\mathrm{\partial <!--\; ???\; -->}{\mathbb{R}}_{+}^n\right),}$$ where $f\in <!--\; ???\; -->L^p\left(\mathrm{\partial <!--\; ???\; -->}{\mathbb{R}}_{+}^n\right),g\in <!--\; ???\; -->L^{q^{\prime }}\left({\mathbb{R}}_{+}^n\right)\text{and}p,q^{\prime }\in <!--\; ???\; -->\left(1+\mathrm{\infty <!--\; ???\; -->}\right),2\le <!--\; ???\; -->\mathrm{\alpha <!--\; ??\; -->}<n\text{satisfying}\frac{n-<!--\; ???\; -->1}{np}+\frac{1}{q^{\prime }}+\frac{2-<!--\; ???\; -->\mathrm{\alpha <!--\; ??\; -->}}{n}=1.$Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant C_n,α,p,q′. Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore, in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler-Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ${\mathrm{\xi <!--\; ??\; -->}}_0\in <!--\; ???\; -->\mathrm{\partial <!--\; ???\; -->}{\mathbb{R}}_{+}^n.$ Our results proved in this paper play a crucial role in establishing the Stein-Weiss inequalities with the Poisson kernel in our subsequent paper.

[en] In this paper a problem of constraint stabilization of a two-wheeled sleigh is considered. This problem is solved with the help of the Chaplygin’s approach, in which Lagrange equations of the second kind are modified with respect to the nonholonomic constraints. For the obtained equations we define the functions of reactions forces of constraints with respect to their stabilization. During the numerical integration some of the stabilization parameters are defining at each step of the summation. This gives an advantage in comparison with the classical stabilization approach. (paper)

[en] The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is established, and the fractional Lagrange equations are obtained by virtue of the d'Alembert—Lagrange principle with fractional derivatives. As the Lie symmetry theorem is based on the invariance of differential equations under infinitesimal transformations, by introducing the differential operator of infinitesimal generators, the determining equations are obtained. Furthermore, the limit equations, the additional restriction equations, the structural equations, and the conserved quantity of Lie symmetry are acquired. An example is presented to illustrate the application of results. (general)

[en] In this work, we study the Willmore submanifolds in a closed connected Riemannian manifold which are orbits for the isometric action of a compact connected Lie group. We call them homogeneous Willmore submanifolds or Willmore orbits. The criteria for these special Willmore submanifolds is much easier than the general theory which requires a complicated Euler-Lagrange equation. Our main theorem claims, when the orbit type stratification for the group action satisfies certain conditions, then we can find a Willmore orbit in each stratified subset. Some classical examples of special importance, like Willmore torus, Veronese surface, etc., can be interpreted as Willmore orbits and easily verified with our method. Our theorems provide a large number of new examples for Willmore submanifolds, as well as estimates for their numbers which are sharp in some classical cases.

[en] The energy functional of Thomas-Fermi-Dirac-von Weizsäcker model with external potential is studied. The minimizer for the functional is investigated. Furthermore, the value and some properties of the minimizer are estimated from the functional without solving the associate Euler-Lagrange equation

[en] For a wide class of Lagrangian systems it is shown rigorously that the conventional formulation of Noether's theorem provides a bijective map from the set of equivalence classes of Noether's symmetries onto the set of equivalence classes of conserved currents. The author further discusses if Noether's theorem is generalized in a significant way by several formulations proposed in this decade. (Auth.)

No abstract available

[en] In this paper, we present a new method to construct the conservation laws for relativistic mechanical systems by finding corresponding integrating factors. First, the Lagrange equations of relativistic mechanical systems are established, and the definition of integrating factors of the systems is given; second, the necessary conditions for the existence of conserved quantities of the relativistic mechanical systems are studied in detail, and the relation between the conservation laws and the integrating factors of the systems is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the systems are established, and an example is given to illustrate the application of the results.