arXiv subjects

Marián Fecko

Publications and source records attributed to Marián Fecko.

4 recordsLinked to original sources

arXiv2024

Galilean and Carrollian Hodge star operators

The standard Hodge star operator is naturally associated with metric tensor (and orientation). It is routinely used to concisely write down physics equations on, say, Lorentzian spacetimes. On Galilean (Carrollian) spacetimes, there is no canonical (nonsingular) metric tensor available. So, the usual construction of the Hodge star does not work. Here we propose analogs of the Hodge star operator on Galilean (Carrollian) spacetimes. They may be used to write down important physics equations, e.g. equations of Galilean (Carrollian) electrodynamics.

math-ph↗

arXiv2023

Some useful operators on differential forms in Galilean and Carrollian spacetimes

Differential forms on Lorentzian spacetimes is a well-established subject. On Galilean and Carrollian spacetimes it does not seem to be quite so. This may be due to the absence of Hodge star operator. There are, however, potentially useful analogs of Hodge star operator also on the last two spacetimes, namely intertwining operators between corresponding representations on forms. Their use could perhaps make differential forms as attractive tool for physics on Galilean and Carrollian spacetimes as forms on Lorentzian spacetimes definitely proved to be.

math-ph↗

arXiv2022

Vector calculus in two-dimensional space

Vector calculus in three-dimensional space is ubiquitous in applications of mathematics in physics and engineering. Its two-dimensional version is, however, quite rare. Here we try to provide a pedagogical account of the subject. It is based on the logic of theory of differential forms. For readers not familiar with the latter, the results are presented in detail in the standard language and notations.

math.HO↗

arXiv2013

On p-form vortex-lines equations on extended phase space

In differential-geometric language, vortex-lines equations on extended phase space of a system may be written as $i_{\dot γ}dσ=0$, where $σ$ is a differential 1-form. This is the structure, to give a paradigmatic example, of the Hamilton equations. Here, we study equations of the same structure, where $σ$ is a differential p-form.

math-ph↗