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Marian Fecko

Publications and source records attributed to Marian Fecko.

6 recordsLinked to original sources
arXiv2017

A generalization of vortex lines

Helmholtz theorem states that, in ideal fluid, vortex lines move with the fluid. Another Helmholtz theorem adds that strength of a vortex tube is constant along the tube. The lines may be regarded as integral surfaces of a 1-dimensional integrable distribution (given by the vorticity 2-form). In general setting of theory of integral invariants, due to Poincare and Cartan, one can find $d$-dimensional integrable distribution whose integral surfaces show both properties of vortex lines: they move with (abstract) fluid and, for appropriate generalization of vortex tube, strength of the latter is constant along the tube.

math-ph↗
arXiv2014

Modern geometry in not-so-high echelons of physics: Case studies

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.

physics.flu-dyn↗
arXiv2013

On symmetries and conserved quantities in Nambu mechanics

In Hamiltonian mechanics, a (continuous) symmetry leads to conserved quantity, which is a function on (extended) phase space. In Nambu mechanics, a straightforward consequence of symmetry is just a relative integral invariant, a differential form which only upon integration over a cycle provides a conserved real number. The origin of the difference may be traced back to a shift in degrees of relevant forms present in equations of motion, or, alternatively, to a corresponding shift in degrees of relevant objects in action integral for Nambu mechanics.

math-ph↗
arXiv1997

On 3+1 decompositions with respect to an observer field via differential forms

3+1 decompositions of differential forms on a Lorentzian manifold (M,g;+ - - -) with respect to arbitrary observer field and the decomposition of the standard operations acting on them are studied, making use of the ideas of the theory of connections on principal bundles. Simple explicit general formulas are given as well as their application to the Maxwell equations.

gr-qc↗
arXiv1997

"Falling cat" connections and the momentum map

We consider a standard symplectic dynamics on TM generated by a natural Lagrangian L. The Lagrangian is assumed to be invariant with respect to the action TR_g of a Lie group G lifted from the free and proper action R_g of G on M. It is shown that under these conditions a connection on principal bundle pi: M \rightarrow M/G can be constructed based on the momentum map corresponding to the action TR_g. The horizontal motion is shown to be in physical terms the one with all the momenta corresponding to the symmetry vanishing. A simple explicit formula for the connection form is given. For the special case of the standard action of G = SO(3) on M = R^3 x ... x R^3 corresponding to a rigid rotation of a N-particle system the formula obtained earlier by Guichardet and Shapere/Wilczek is reproduced.

math-ph↗