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Published in:
2019 | OriginalPaper | Chapter
Integral Invariants (Poincaré–Cartan) and Hydrodynamics
Author : Marián Fecko
Published in: Geometric Methods in Physics XXXVI
Publisher: Springer International Publishing
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There are several ways how hydrodynamics of ideal fluid may be treated geometrically. In particular, it may be viewed as an application of the theory of integral invariants due to Poincaré and Cartan (see Refs. [1, 2], or, in modern presentation, Refs. [3, 4]). Then, the original Poincaré version of the theory refers to the stationary (time-independent) flow, described by the stationary Euler equation, whereas Cartan’s extension embodies the full, possibly time-dependent, situation.Although the approach via integral invariants is far from being the best known, it has some nice features which, hopefully, make it worth spending some time. Namely, the form in which the Euler equation is expressed in this approach, turns out to be ideally suited for extracting important (and useful) classical consequences of the equations remarkably easily (see more details in Ref. [4]). This refers, in particular, to the behavior of vortex lines, discovered long ago by Helmholtz.