DIFFERENTIAL GEOMETRY, CONTINUUM MECHANICS AND MATHEMATICAL PHYSICS Seminar

 

PROGRAM

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Predavanja možete pratiti i online putem MITEAM stranice seminara Diferencijalna geometrija, mehanika kontinuuma i matematička fizika: https://miteam.mi.sanu.ac.rs/asset/YNvstaSeP9v2tgqTP

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Plan rada seminara Diferencijalna geometrija, mehanika kontinuuma i matematička fizika za MAJ 2025.

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Petak, 16.05.2025. u 16:15, Pariske Komune bb, Niš i Live stream
Nikola Nešić, Faculty of Technical Sciences, University of Priština in Kosovska Mitrovica
ROBOTICS BASED ON SCREW THEORY, MATRIX EXPONENTIALS AND LIE ALGEBRAS
The foundation of the screw theory approach can be traced back to the Mozzi-Chasles theorem, which describes rigid-body motion as a combination of rotation and translation along a screw axis. A significant breakthrough in making classical screw theory more accessible occurred in the early 1980s when Roger Brockett demonstrated how to mathematically represent kinematic chains using the Lie group structure of rigid-body motions. This insight enabled the reinterpretation of screw theory through fundamental concepts of linear algebra and differential equations. With this "modern screw theory", the advanced tools of differential geometry can be effectively applied to a broad range of robotics challenges, which will be discussed in this prezentation.

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Petak, 23.05.2025. u 12:15, Pariske Komune bb, Niš i Live stream
Katica Hedrih, Mathematical Institute of the SASA
FIVE THEOREMS ON ORTHOGONAL, CURVILINEAR SYSTEMS OVER REVOLVING SURFACES AND TEN THE­O­REMS ON THE PROPERTIES OF THE DYNAMICS OF ROLLING A HEAVY BALL ON REVOLVING SURFACES
The construction of a series of curvilinear orthogonal coordinate systems over a series of revolving surfaces is presented. The geometry of coordinate surfaces and coordinate lines, of thus constructed curvilinear orthogonal coordinate systems over a set of revolving surfaces with the corresponding basis vectors and matrices of metric tensors, will be presented. Five theorems on the properties of the elements of such curvilinear orthogonal coordinate systems are formulated. Some of these theorems were presented at the world conference NODYCON Rome 2023, at La Sapienza University, as well as at the large All-Russian Congress of Theoretical and Applied Mechanics in Ufa 2019. Then, nonlinear differential equations and elements of nonlinear dynamics of rolling without slipping of a heavy ball on revolving surfaces are presented. Analytical expressions of the corresponding components of the angular velocities of rolling are determined, and the directions of the instantaneous axes of the component rolls are indicated. The studied nonlinear dynamics of rolling on revolving surfaces allowed the definition of ten theorems on the properties of this rolling dynamics. Based on the scientific results of these studies, a large number of papers have been published, of which we list the following three here:

1. K. R. (Stevanović) Hedrih, Rolling heavy ball over the sphere in real Rn3 space, Nonlinear Dyn. 97 (2019), 63-82.
2. K. R. (Stevanović) Hedrih, Rolling a heavy ball on a revolving surface, Innovative Mechanical EngineEring 4(1) (2025).
3. K. R. (Stevanović) Hedrih, An Overview: Rolling of a Heavy Ball on Curvilinear Paths and Surfaces of the Basis of Nonlinear Dynamics of Radial and Spherical Ball Bearings in Machine Systems, In: J. M. Balthazar, P. B. Gonçalves, A. M. Tusset, G. Litak, J. Simonovic (eds), Nonlinear Dynamics, Chaos, Control, Energy Transfer and Their Applications in Engineering Sciences, DYCAELS 2023, Mechan. Machine Science 142, 2025, Ch. 8.

Then, the lecture presents applications of the previous results to the nonlinear dynamics of rolling a heavy ball on a number of characteristic revolving surfaces: sphere, torus, cone, as well as on parabolic revolving surfaces and biquadratic parabola revolving surfaces. The corresponding nonlinear differential equations and cyclic integrals are presented, which describe nonlinear dynamics. The importance of choosing generalized coordinates, which can easily solve the problems posed in this way, is pointed out. The importance of the cyclic coordinate and the cyclic integral is also pointed out. The first integrals of the corresponding nonlinear differential equations are also presented.
Zajednički sastanak sa Odeljenjem za mehaniku.

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