Abstract: Given a wing, a weight, and a fixed supply of energy, how should the wing move in air so that it can transport the weight the furthest or lift the weight for the longest? Should the wing flap like birds and insects or cuts through the air like an airfoil? To fly from point A to point B, a fixed wing travels straight, but a flapping wing traverses up and down, which seems to waste energy. On the other hand, a flapping wing has infinitely more possible motions, and no theorem predicts that all of them are worse than a fixed wing motion. How to find some of the possibly efficient flapping motions in this infinite parameter space? I will take a step to answer these questions.

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