  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. Finally, systems fully constrained by direct nonholonomic
constraints (e.g., passive wheels) can push against these constraints to move
through the world. 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 in a short time, 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 effort-based 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 effort 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.
