Understanding the terms linear and nonlinear transformation of coordinates and coordinate systems and their properties is the basis for studying the nonlinear dynamics of systems. In this paper, we focus our attention on two generalized functional coordinate systems of curvilinear coordinates: generalized elliptical curvilinear coordinate system and generalized spheroidal curvilinear coordinate system. In these functional coordinate systems of curvilinear coordinates, we will determine the base vectors of the tangent space of the kinetic point position vector of a material system and determine their changes with time and the motion of the kinetic point in real space. The tangent space of the position vector of the kinetic point of the observed material system is deformed by three base vectors in the functional curvilinear coordinate system. Also, the changes of the base vectors of that tangent space of the position vector of the kinetic point of the observed material system are described. The base vectors rotate at certain angular velocities and their intensities change with time and the motion of the kinetic point. In this paper we present analytical expressions of angular velocities of rotation of base vectors of tangent space of vector of position of kinetic point of observed material system in two generalized functional coordinate systems of curvilinear coordinates: generalized elliptical curvilinear coordinate system and generalized oblate spheroidal curvilinear coordinare system. We will also present analytical expressions of the relative change in the intensity of these basic vectors in both coordinate systems. We will also point out the analogy of the base vectors of the tangent space of the position vector with the model of defotmation of line elements of the elementary part of a deformable body in the process of deformation under the action of forces, when their lengths and positions change by extension and rotation.
Keywords: Linear, nonlinear, transformation, curvilinear coordinate systems, basic vectors, vector position, tangent space, angular velocity of basic vector rotation, velocity of basic vector dilatation, generalized elliptical curvilinear orthogonal coordinate system, generalized oblate spheroidal curvilinear orthogonal coordinare system.
References
[1] Hedrih (Stevanović) R. Katica, (2021), Linear and non-linear transformation of coordinates and angular velocity and intensity change of basic vectors of tangent space of a position vector of a material system kinetic point, The European Physical Journal Special Topics, (accepted 8 ju;y 2021- EPJS-D-21-00010R1) DOI: 10.1140/epjs/s11734-021-00226-6 ; ISSN: 1951-6355
[2] Hedrih (Stevanović) R. Katica , (2010), Visibility or appearance of nonlinearity, Tensor, N.S. Vol. 72, No. 1 (2010), pp. 14-33, #3. Tensor Society, Chigasaki, Japan, ISSN 0040-3504.
[3] Hedrih (Stevanović) K., (2012), Tangent space extension of the position vectors of a discrete rheonomic mechanical system, Professor N. R. Sen Memorial Lecture, Bulletin of the Calcutta Mathematical Society Volume 104, No.2(2012) pp. 81-102. Bull.Cal.Math. 104 (2) 81-102 (2012).
[4] Hedrih (Stevanović) K., (2014), Angular velocityand intensity under change of basic vectors of position vector of tangent space of a meterial system kinetic point –Consideration of the difference betwenn linear and nonlinear transformations, To memory of academician Vladimir Metodievich Matrosov (May 8, 1932-April 17,2011) President of Academy of nonlinear Sciences. Tensor, Vol. 75, No. 1 pp. 71-93. Tensor Society (Tokyo), c/o Kawaguchi Inst. of Math. Soc. , Japan.
[5] Heinbockel J.H., Introduction to Tensor Calculus and Continuum Mechanics, Department of Mathematics and Statistic Old Dominion University, Copyright c 1996 by J.H. Heinbockel. All rights reserved. Reproduction and distribution of these notes is allowable provided it is for non-profit purposes only.