DIFFERENTIAL GEOMETRY, CONTINUUM MECHANICS AND MATHEMATICAL PHYSICS Seminar

 

PROGRAM

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Plan rada Seminara diferencijalna geometrija, mehanika kontinuuma i matematička fizika za MAJ 2024.

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Petak, 17.05.2024. u 12:15, Regionalni centar Niš, Pariske Komune bb, Niš Online
Katica (Stevanović) Hedrih, Mathematical Institute SANU, Belgrade, Serbia
AN OVERVIEW: WITH THE ANDJELIĆ AND RAŠKOVIĆ TENSORS INTO TRANSFORMATION OF BASE VECTORS OF TANGENT SPACE OF THE POSITION VECTOR OF THE KINETIC POINTS, VECTOR ROTATION OF LINE ELEMENT OF DEFORMABLE BODY, AND ELEMENTS OF MIHAILO PETROVIĆ'S MATHEMATICAL PHENOMENOLOGY WITH ANALOGIES OF MATRICES OF STRESS TENSORS, STRAIN TENSORS, AND VECTORS OF MASS MOMENTS
Starting from the notion of linear and nonlinear transformations, affine and functional-nonlinear mappings of coordinates and coordinate systems, geometrical and kinematical invariants along linear or nonlinear transformations, the coordinates from one coordinate system to another are pointed out. In a curvilinear coordinate system, the coordinates of a geometrical or kinematical point are not equal to the coordinates of its corresponding position vector. Expressions of basic vectors of the tangent space of kinetic point vector positions in generalized curvilinear coordinate systems for the cases of orthogonal and nonorthogonal curvilinear coordinate systems are derived. Examples of expressions of basic vectors of the tangent space of kinetic point vector positions in polar-cylindrical, spherical, parabolic-cylindrical, and three-dimensional-three-parabolic system of curvilinear orthogonal coordinates, as well as the revolving parabolic orthogonal curvilinear system, are presented. Next, expressions of the change of basic vectors of the tangent space of kinetic point vector positions with time are also provided. Geometrical (physical), covariant, and contravariant coordinates of the position vector of a kinetic mass point in a coordinate system determined by basic vectors of the tangent space of this kinetic point vector position in generalized curvilinear coordinate systems are pointed out and determined. Original expressions of angular velocity and velocity of dilatations of basic vectors of the tangent space of kinetic point vector position in generalized curvilinear coordinate systems, as well as in a series of special orthogonal curvilinear coordinate systems, are derived by the lecturer of this lecture and will be presented. Also, the lecture will present the rotation vector of the line element of the deformed deformable body in the domain of elastic deformation. Elements of Mihailo Petrović's mathematical phenomenology and analogies of matrices of stress tensors, strain tensors, and state mass moments using vectors of mass moments will be presented. The mathematical analogies between vector models of stress state vector model, strain state vector model, and mass inertia moment state vector model will also be presented. The contents of this lecture are composed on the basis of the lecturer’s published papers in the following journals: Springer's journal European Journal of Physics-Special Topics (2021), the Tensor journal of the Japanese Tensor Society of the same name (1997), Facta Universitatis Series Mechanics, Automatic Control and Robotics, and Journal of MAI Moscow, and the accepted Abstract for ICTAM Daegu Corea 2024, and others.

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