Closed Catenary Loops: Lariat Chain, String Shooter, and Heavy Elastica

We consider the equilibrium configurations of a closed loop of axially-moving material subject to uniform body and tangential drag forces, and one external point force, with allowance for a jump in angle at that point. This is an idealized model of a thin body driven through air by a small wheel or pair of wheels. We will review the history of this and closely related problems, going back to the 19th century, provide a general perspective and selection of solutions, and address several errors and misconceptions in the literature. As the ratio of drag forces to weight increases, analytical solutions for a perfectly-flexible string undergo two bifurcations, separating three regimes with distinctly different shapes and singular behaviors. In the lowest-drag regime, closed loop solutions require the addition of a small amount of bending stiffness, but this regularization has an atypical form. We will also discuss the global balances of linear, angular, and pseudo-momentum; the latter is further related to energy losses through drag.