  Isolated mechanical systems -- e.g., those floating in space, in free-fall,
or on a frictionless surface -- are able to achieve net rotation by cyclically
changing their shape, even if they have no net angular momentum. Similarly,
swimmers immersed in "perfect fluids" are able to use cyclic shape changes to
both translate and rotate even if the swimmer-fluid system has no net linear or
angular momentum. Previous work has demonstrated that the net displacement
induced by these shape changes corresponds to the amount of /constraint
curvature/ that the gaits enclose. To properly assess or optimize the utility
of a gait, however, we must also consider the time or resources required to
execute it: A gait that produces a small displacement per cycle, but that can
be executed at high frequency, may produce a faster average velocity than a
gait that produces a large displacement per cycle, but takes much longer to
complete a cycle at the same average instantaneous effort. In this paper, we
consider two energetic cost functions for assessing the costs associated with
executing these cycles. For each of these cost functions, we demonstrate that
fixing the average instantaneous cost to a unit value allows us to transform
the energetic costs into time-to-execute costs for any given gait cycle. We
then illustrate how the interaction between the constraint curvature and these
costs leads to characteristic geometries for optimal cycles, in which the gait
trajectories resemble elastic hoops distended from within by internal
pressures.
