Coaxial axisymmetric vortex rings: 150 years after Helmholtz

This article addresses the fascinating 150 years history of the classical Helmholtz paper that laid the foundation of the vortex dynamics. Among general theorems on vortex motion, this memoir contains the special section on circular vortex filaments and axisymmetric vortex rings, in particular. The objective of this article is both to clarify some purely mathematical questions connected with the Dyson model of coaxial vortex rings in inviscid incompressible fluid and to provide a historical overview of achievements in experimental, analytical, and numerical studies of vortex rings interactions. The model is illustrated by several examples both of regular and chaotic motion of several vortex rings in an unbounded fluid.