Variations on a theme by Cauchy Uriel Frisch Observatoire de la Côte d'Azur Cauchy's invariants for 3D incompressible Euler flow, recently rediscovered after 200 years, give us a powerful tool for investigating the Lagrangian structure of such flow. Among the topics we shall discuss: how the Cauchy invariants relate to the Lie-transport (pullback) invariance of the vorticity 2-form; how they can be generalized to higher-order forms and to Euler flow on Riemann manifolds; how they generate recursion relations giving a constructive hold on time-analyticity of the Lagrangian map and thereby allow the development of Cauchy-Lagrange numerical schemes that can be orders of magnitude faster than the usual Eulerian schemes. Various open problems will be mentioned, e.g. blow-up in wall-bounded flow. Some useful references: * 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].