Closing a catenary loop: the lariat chain, the string shooter, and the heavy elastica

We review, critique, and extend results related to the problem of closed loop shape equilibria of a string shooter, a type of catenary consisting of steady, axially moving configurations of an inertial, inextensible, perfectly flexible string in the presence of gravity and drag forces. We relate to similar problems, including the lariat (no gravity), chain fountain (not closed), and heavy \emph{elastica} (bending stiffness). We focus on the difficulty inherent to continuing a catenary through a vertical orientation, necessary to close a loop, which difficulty changes in nature as the system undergoes bifurcations with increasing drag. We construct solutions by implementing available analytical results, and numerically generate additional solutions with added bending stiffness. We briefly discuss global balances of linear, angular, and pseudo-momentum for this system.