2014/02/28 by Uriel Frisch, Barbara Villone · 62 citations
Earth and Planetary Sciences · Engineering · Mathematics · Physics and Astronomy · #Cauchy distribution #Cauchy problem #Compressibility #Euler equations #Euler's formula #Flow (mathematics) #Fluid dynamics and aerodynamics studies #Lagrangian and Eulerian specification of the flow field #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing #Vortex #Vorticity #math.AP #math.HO #physics.hist-ph
paper · pdf · doi:10.1140/epjh/e2014-50016-6
published in The European Physical Journal H 39(3), 325-351 (Springer Science+Business Media) · 23 pages, 6 figures, EPJ H (history), in press
openalex publication_date 2014/09/01 · arxiv created 2014/09/12 · arxiv updated 2014/09/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to Göttingen University, contain major discoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal incompressible flow in terms of three invariants that generalize to three dimensions the now well-known law of conservation of vorticity along fluid particle trajectories for two-dimensional flow. This has very recently been used to prove analyticity in time of fluid particle trajectories for 3D incompressible Euler flow and can be extended to compressible flow, in particular to cosmological dark matter. Hankel showed that Cauchy's formulation gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux invariants and, in the middle of the proof, derived an intermediate result which is the conservation of the circulation of the velocity around a closed contour moving with the fluid. This circulation theorem was to be rediscovered independently by William Thomson (Kelvin) in 1869. Cauchy's invariants were only occasionally cited in the 19th century --- besides Hankel, foremost by George Stokes and Maurice Lévy --- and even less so in the 20th until they were rediscovered via Emmy Noether's theorem in the late 1960, but reattributed to Cauchy only at the end of the 20th century by Russian scientists.