Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow and modern perspectives

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 in fluid dynamics and cosmology is only beginning to be felt.

Cauchy found a Lagrangian formulation of 3D ideal incompressible flow in terms of three invariants, which generalizes to three dimensions the law of conservation of vorticity along fluid particle trajectories for two-dimensional flow. The very concept of invariant, one hundred years before the work of Emmy Noether relating invariants and continuous symmetries, was foreign to early 19th century thinking. This probably explains why Cauchy’s invariants were quickly forgotten by almost everybody. Actually, only a corollary of the invariants, called the Cauchy vorticity formula, remained well-known.

Hankel showed that the Cauchy invariants 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 actually 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 Levy – and even less so in the 20th until they were rediscovered via Noether’s theorem in the late 1960, but reattributed to Cauchy only at the end of the 20th century by Russian scientists.

Actually, Cauchy’s Lagrangian formulation, combined with a technique of recursion relations for time-Taylor coefficients, first used by cosmologists in the 1990s, gives a powerful tool for tackling issues of time-analyticity of fluid particle trajectories and of numerical integration in Lagrangian coordinates. Such techniques apply not only to the incompressible Euler equations (with or without boundaries) but to other equations possessing similar invariants. This includes the Euler-Poisson equation for self-gravitating dark matter in an Einstein-de Sitter or Lambda CDM Universe.

* Frisch, U. and Villone, B. 2014. Cauchy’s almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow, Europ. Phys. J. H 39, 325–351. arXiv:1402.4957 [math.HO]
* Frisch, U. and Zheligovsky, V. 2014. A very smooth ride in a rough sea, Commun. Math. Phys., 326, 499–505. arXiv:1212.4333 [math.AP]
* Zheligovsky, V. and Frisch, U. 2014. Time-analyticity of Lagrangian particle trajectories in ideal fluid flow, J. Fluid Mech., 749, 404–430. arXiv:1312.6320 [math.AP]
* Rampf, C., Villone, B and Frisch, U. 2015. How smooth are particle trajectories in a Lambda-CDM Universe?, Mon. Not. R. Astron. Soc., 452}, 1421–1436. arXiv:1504.00032 [astro-ph.CO]
* Podvigina, O., Zheligovsky, V. and Frisch, U. 2015. The Cauchy-Lagrangian method for numerical analysis of Euler flow. J. Comput. Phys., 306, 320–342. arXiv:1504.05030v1 [math.NA]
* Besse, N. and Frisch, U. A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain, submitted. arXiv:1603.09219v1 [math.AP]