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Acta Physica Slovaca 63, No.5, 261 – 359 (2013) (99 pages)
MODERN GEOMETRY IN NOT-SO-HIGH ECHELONS OF PHYSICS: CASE STUDIES Marián Fecko Department of Theoretical Physics and Didactics of Physics, Comenius University, Bratislava, Slovakia Full text: ::pdf :: (Received 4 March 2014, accepted 14 May 2014) Abstract: In this mostly pedagogical tutorial article a brief introduction to modern geometrical treatment of fluid dynamics and electrodynamics is provided. The main technical tool is standard theory of differential forms. In fluid dynamics, the approach is based on general theory of integral invariants (due to Poincare and Cartan). Since this stuff is still not considered common knowledge, the first chapter is devoted to an introductory and self-contained exposition of both Poincare version as well as Cartan’s extension of the theory. The main emphasis in fluid dynamics part of the text is on explaining basic classical results on vorticity phenomenon (vortex lines, vortex filaments etc.) in ideal fluid. In electrodynamics part, we stress the aspect of how different (in particular, rotating) observers perceive the same space-time situation. Suitable 3 + 1 decomposition technique of differential forms proves to be useful for that. As a representative (an simple) example we analyze Faraday’s law of induction (and explicitly compute the induced voltage) from this point of view. DOI: 10.2478/apsrt-2013-0005 PACS: 02.40.-k, 47.10.A-, 47.32.-y, 03.50.De Keywords: Ideal fluid, barotropic flow, vortex lines, transport theorem, Helmholtz theorem, lines of solenoidal field, integral invariant, 3+1 decomposition, rotating frame, Faraday’s law |
ISSN 1336-040X (online) ISSN 0323-0465 (printed) For authors: ActaStyle.cls |
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