The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics

Abstract

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 throughout this century. After a partial re-discovery (Ricca, 1991a), new material has now been found and is presented here with a full review of their work in relation to the present state of the art in non-linear mechanics and vortex dynamics. Their results include the conception of the localized induction approximation (LIA) for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of motion, the asymptotic potential theory applied to vortex tubes, the derivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations and the stability analysis of circular vortex filaments. In the light of modern developments in non-linear fluid mechanics, their work strikes for modernity and depth of results. Even more striking is the fact that this work remained obscure for almost a century. The results of Da Rios are particularly important in the study of integrable one-dimensional systems and vortex filament motion; Levi-Civita's work on asymptotic potential for slender tubes is at the core of the mathematical formulation of potential theory and capacity theory.