×

Integral invariants (Poincaré-Cartan) and hydrodynamics. (English) Zbl 1426.53101

Kielanowski, Piotr (ed.) et al., Geometric methods in physics XXXVI. Workshop and summer school, Białowieża, Poland, July 2–8, 2017. Selected papers of the 36th workshop (WGMPXXXVI) and extended abstracts of lectures given at the 6th “School of geometry and physics”. Cham: Birkhäuser. Trends Math., 377-382 (2019).
Summary: 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 [H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Tome III. Invariants intégraux. Solutions périodiques du deuxième genre. Solutions doublement asymptotiques. Paris: Gauthier-Villars et Fils (1899; JFM 30.0834.08); E. Cartan, Leçons sur les invariants intégraux. Cours professé à la Faculté des Sciences de Paris. Paris: A (1922; JFM 48.0538.02)], or, in modern presentation [V. I. Arnold, Mathematical methods of classical mechanics. Translated by K. Vogtman and A. Weinstein. New York, NY: Springer (1978; Zbl 0386.70001); the author, “Modern geometry in not-so-high echelons of physics: case studies”, Acta Phys. Slovaca 63, No. 5, 261–359 (2013)]). 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 [the author, loc. cit.]). This refers, in particular, to the behavior of vortex lines, discovered long ago by Helmholtz.
For the entire collection see [Zbl 1417.53002].

MSC:

53Z05 Applications of differential geometry to physics
76B47 Vortex flows for incompressible inviscid fluids
PDFBibTeX XMLCite
Full Text: DOI