This question for the case of a homogeneous field—the gravity field—was discussed in detail by Prof. M.I. Akimov, who found all surfaces that allow the movement of a heavy point along helical lines with a vertical axis in the presence of friction, and investigated the conditions for these surfaces under which the movement in question would be stable.
Let FT and Fn be the absolute values of the tangent and normal to the surface components of the weight force acting on a point. A point placed on a surface without an initial velocity” remains in equilibrium in the presence of friction if Fr~kFN> and begins to move if Fr>kFN, where k is the friction coefficient. Differential equations of motion on a rough surface (g, tp, z) = 0 point M of weight g in cylindrical coordinates 1 g, 2 (z axis is directed vertically downwards) - see. article.
When studying the steady motion of a heavy material particle along a helical line on a rough helical surface with the Z axis of the cylindrical coordinate system directed vertically upward, M. I. Akimov noted helical surfaces corresponding to the values of the function (see article), the movement of a heavy point along helical lines, in the presence of friction, has special properties. On the first of these surfaces, the movement in question (with appropriate friction coefficients) along all helical lines is possible only at the same constant speed. On the second surface (with appropriate friction coefficients) along all helical lines it is possible only with the same constant vertical component of speed. Such a surface can find application in those designs of a spiral separator when the unloading of dissimilar separated particles of the mixture must occur simultaneously. On the third surface, with the same coefficient of friction k, the steady motion of heavy particles can occur along any helical line. This work contains the calculations necessary to construct the three helical surfaces under consideration for the purpose of experimentally studying the steady-state movements of heavy particles occurring on them.
Generalizing the Catalan problem about the motion of a point rotating on a polished surface, the author searches for surfaces that allow the motion of a point rotating in a spiral with a vertical axis in the presence of friction, and explores the nature of the stability of the motion of a point of rotation on a spiral located on a rough helical surface with a vertical axis, and also the influence on this movement of the resistance of the environment, depending on the speed, and the rotation of the surface around the aerodynamic propellers.
In this work, the elementary conclusions are presented and a number of new surfaces are indicated that can find application in the theory of a spiral separator and represent, with proper selection, known advantages over an ordinary oblique helical surface, both in the sense of achieving greater stability of the screw movement occurring on them, and and, if possible, an appropriate choice of the distribution of velocities of the masses moving along them. This work also presents new results related to the study of the stability of the considered movement along a helical line on a rough surface, and the influence on this movement is clarified: 1) the resistance of the medium, 2) the rotation of the surface around the vertical axis of the helical lines located on it.
The purpose of this note is to simplify the method proposed by academician N. M. Krylov for constructing a given function in the form of a series acting in Jacobi polynomials. By forming an a priori expansion with uniform and absolute convergence, we show, without relying on the Riesz-Fischer theorem, that it can be identified with expansions in Jacobi polynomials.
Chapter IV. Some series expansions in generalized Bessel's functions. Chapter V Application of generalized Bessel's functions to problems of mechanics.
The subject of this work is the study of functions that occur initially in the form of a definite integral (see article). As a simple example to explain the application of the obtained formulas, I give the classical problem of the motion of a spherical pendulum in the case of its small oscillations around the lowest equilibrium position. Examples of problems leading to the generalized Kepler's equation. The origin of Bessel's functions of many variables and their expression in the form of infinite series (see article). The author pays attention to equations satisfied by generalized Bessel's functions and the general solution of these equations.
A detailed description of the first proof of Gauss's fundamental theorem and Cauchy's proof is given (see article).
Let us investigate the limiting cases of the Riemann function. Let us take the differential equation P of the function and consider the functions (see article). Following Riemann's example, we imagine the path of integration in the form of a flexible, stretchable and easily movable thread. As it moves, the special point deforms this path, pushing it in front of itself and never crossing it. With this representation of the integration path from closed curves corresponding to integrals (11), (13), (15), for integrals (16), (18), (20) we obtain open paths in certain directions extending to infinity.
