Limits on Propulsion from Rest in Fluids

Propulsion from rest in fluids, whether in a rowing stroke pushing through essentially still water or the explosive kick of a frog, involves a tradeoff between impulse and work. The agent has a fixed work budget with which they can increase their momentum in a target direction. Thus, it is natural to ask, what is the greatest impulse that they can exert for a fixed amount of work, and how can this be achieved.

To generalize this problem, we abstract the propelling object as a body force acting on a bounded region over a fixed time and construct the following optimization problem.

Given an ideal, incompressible fluid occupying R2 or R3, a bounded region A, a time limit T, and a work budget W, find the spatiotemporal body‑force distribution supported in A that maximizes net impulse in a prescribed direction.

We solve this using a variational approach. In the limit of T approaching zero, we obtain a closed form for the maximum impulse. For finite T, we use steepest descent to compute optimal force histories.

Our results establish theoretical bounds on the impulse–work trade‑off and suggest design principles in both engineered propulsors and organisms that swim via impulsive strokes.