THE CONTRIBUTIONS OF DA RIOS AND LEVI-CIVITA TO ASYMPTOTIC POTENTIAL-THEORY AND VORTEX FILAMENT DYNAMICS

In this paper we present for the first time a detailed account of the work of L.S. Da Rios and T. Levi-Civita on what is believed to be one of the first major contributions to three-dimensional vortex filament dynamics, Their work spanned a period of almost 30 years, from 1906 to 1933, and despite many publications remained almost unnoticed through out this century. After a partial re-discovery (Ricca, 1991a), new mat erial has now been found and is presented here with a full review of t heir work in relation to the present state of the art in non-linear me chanics and vortex dynamics. Their results include the conception of t he localized induction approximation (LIA) for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of m otion, the asymptotic potential theory applied to vortex tubes, the de rivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations and the stability analysis of cir cular vortex filaments. In the light of modern developments in non-lin ear fluid mechanics, their work strikes for modernity and depth of res ults. Even more striking is the fact that this work remained obscure f or almost a century. The results of Da Rios are particularly important in the study of integrable one-dimensional systems and vortex filamen t motion; Levi-Civita's work on asymptotic potential for slender tubes is at the core of the mathematical formulation of potential theory an d capacity theory.