scholarly journals THE BRACHISTOCHRONIC MOVEMENT OF A MATERIAL POINT IN THE HORIZONTAL VECTOR FIELD OF A MOBILE FLUID

2019 ◽  
Vol 0 (3) ◽  
pp. 44-51
Author(s):  
Viktor P. Legeza ◽  
Oleksiy V. Atamaniuk
Keyword(s):  
Vector Field ◽  
Material Point ◽  
Horizontal Vector

Anti-invariant Riemannian submersions from Sasakian manifolds with totally umbilical fibers

2021 ◽  
pp. 2150169
Author(s):  
Gizem Köprülü ◽  
Bayram Şahin
Keyword(s):  
Vector Field ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Sasakian Manifold ◽  
Riemannian Submersion ◽  
Sasakian Manifolds ◽  
Riemannian Submersions ◽  
Totally Umbilical ◽  
Characteristic Vector Field ◽  
Horizontal Vector

The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci curvature of anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds.

DETERMINATION OF THE FASTEST TRAJECTORIES OF MATERIAL POINT MOTION IN A HORIZONTAL VECTOR FIELD

2021 ◽  
Vol 4 ◽  
pp. 19-27
Author(s):  
Victor Legeza ◽  
◽  
Alexander Neshchadym ◽  
Keyword(s):  
Optimal Control ◽  
Vector Field ◽  
Variational Problem ◽  
Material Point ◽  
Minimum Time ◽  
Algebraic Equations ◽  
Starting Point ◽  
Point Motion ◽  
Oscillatory Character ◽  
Pass Through

The article proposes a solution to the well-known Zermelo navigation problem by classical variational methods. The classical Zermelo problem within the framework of optimal control theory is formulated as follows. The ship must pass through the region of strong currents, the magnitude and direction of the current velocity are set as functions of phase variables. In this case, the relative speed of the ship is set, the module of which remains constant during movement. It is necessary to find such an optimal control that ensures the arrival of the ship at a given point in the minimum time, i.e. control of the ship by fast-action should be determined. In this paper, we consider the brachistochronic motion of a material point in a plane vector field of a mobile fluid, for which the classical variational problem of finding extreme trajectories is formulated. The aim of the study is to obtain equations of extreme trajectories along which a material point moves from a given starting point to a given finish point in the least amount of time. The solution to the problem was carried out using the classical methods of the theory of the calculus of variations. For a given variant of the boundary conditions, algebraic equations of extremals of motion of a material point were established in the form of segments of a power series. A comparative analysis of the fast-action was carried out both along extreme trajectories and along an alternative path — along a straight line that connects two given start and finish points. Analysis of the results showed that the considered variational problem has two solutions, which differ only in sign. However, only one solution provides the minimum time for moving a material point between two given points. It was also found that the extreme trajectory of the brachistochronic motion of a point is not straight, but has an oscillatory character.

ENTROPY FORM AND THE CONTACT GEOMETRY OF THE MATERIAL POINT MODEL

2012 ◽  
Vol 09 (01) ◽  
pp. 1250013 ◽  
Author(s):  
M. DOLFIN ◽  
M. FRANCAVIGLIA ◽  
S. PRESTON ◽  
L. RESTUCCIA
Keyword(s):  
Vector Field ◽  
Phase Space ◽  
Dynamical Systems ◽  
Conformal Factor ◽  
Material Point ◽  
Point Model ◽  
Geometrical Meaning ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Hamiltonian Dynamical Systems

In this work we investigate a material point model (MP-model) and exploit the geometrical meaning of the "entropy form" introduced by Coleman and Owen. We show that a modification of the thermodynamical phase space (studied and exploited in numerous works) is an appropriate setting for the development of the MP-model in different physical situations. This approach allows to formulate the MP-model and the corresponding entropy form in terms similar to those of homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that admissible processes curves belong to the (extended) constitutive surfaces foliating the extended thermodynamical phase space [Formula: see text] of the model over the space X of basic variables. Extended constitutive surfaces ΣS,κ are described as the Legendre submanifolds ΣS of the space [Formula: see text]shifted by the flow of Reeb vector field. This shift is controlled by the entropy production function κ. We determine which contact Hamiltonian dynamical systems ξK are tangent to the surfaces ΣS,κ, introduce conformally Hamiltonian systems μξK where conformal factor μ characterizes the increase of entropy along the trajectories. These considerations are then illustrated by applying them to the Coleman–Owen model of thermoelastic point.

Defects in Crystals

1968 ◽  
Vol 26 ◽  
pp. 244-245
Author(s):  
Kenneth R. Lawless
Keyword(s):  
Electron Microscope ◽  
Point Defects ◽  
Surface Defects ◽  
Chemical Properties ◽  
Material Point ◽  
Crystal Defects ◽  
Defects In Crystals ◽  
Irradiation Effects ◽  
Normal Lattice ◽  
Line Defects

One of the most important applications of the electron microscope in recent years has been to the observation of defects in crystals. Replica techniques have been widely utilized for many years for the observation of surface defects, but more recently the most striking use of the electron microscope has been for the direct observation of internal defects in crystals, utilizing the transmission of electrons through thin samples.Defects in crystals may be classified basically as point defects, line defects, and planar defects, all of which play an important role in determining the physical or chemical properties of a material. Point defects are of two types, either vacancies where individual atoms are missing from lattice sites, or interstitials where an atom is situated in between normal lattice sites. The so-called point defects most commonly observed are actually aggregates of either vacancies or interstitials. Details of crystal defects of this type are considered in the special session on “Irradiation Effects in Materials” and will not be considered in detail in this session.

Mathematical Model of Flat Surface Machining Inaccuracies due to Elastic Strains of Technological System

2018 ◽  
pp. 1-14 ◽  
Author(s):  
I. I. Kravchenko
Keyword(s):  
Mathematical Model ◽  
Vector Field ◽  
Model Development ◽  
Technological System ◽  
Mutual Arrangement ◽  
Real Surface ◽  
Main Bearing ◽  
Flat Surfaces ◽  
Internal Surfaces ◽  
Elastic Strains

The paper considers the mathematical model development technique to build a vector field of the shape deviations when machining flat surfaces of shell parts on multi-operational machines under conditions of anisotropic rigidity in technological system (TS). The technological system has an anisotropic rigidity, as its elastic strains do not obey the accepted concepts, i.e. the rigidity towards the coordinate axes of the machine is the same, and they occur only towards the external force. The record shows that the diagrams of elastic strains of machine units are substantially different from the circumference. The issues to ensure the specified accuracy require that there should be mathematical models describing kinematic models and physical processes of mechanical machining under conditions of the specific TS. There are such models for external and internal surfaces of rotation [2,3], which are successfully implemented in practice. Flat surfaces (FS) of shell parts (SP) are both assembly and processing datum surfaces. Therefore, on them special stipulations are made regarding deviations of shape and mutual arrangement. The axes of the main bearing holes are coordinated with respect to them. The joints that ensure leak tightness and distributed load on the product part are closed on these surfaces. The paper deals with the analytical construction of the vector field F, which describes with appropriate approximation the real surface obtained as a result of modeling the process of machining flat surfaces (MFS) through face milling under conditions of anisotropic properties.

On the geometry of Killing vector field

2019 ◽  
Vol 2019 (2) ◽  
pp. 62-67
Author(s):  
R.A. Ilyasova
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Killing Vector Field

Some Properties of Para-Sasakian Manifolds

2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Sasakian Manifold ◽  
Subject Classification ◽  
Killing Vector Field ◽  
Sasakian Manifolds ◽  
Geometric Properties ◽  
Paper Author

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

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