Optimizing snake locomotion in the plane

We develop a numerical scheme to determine which planar snake motions are optimal for locomotory efficiency, across a wide range of frictional parameter space. For a large coefficient of transverse friction, we show that retrograde travelling waves are optimal. We give an asymptotic analysis showing that the optimal wave amplitude decays as the - 14 power of the coefficient of transverse friction. This result agrees well with the numerical optima. At the other extreme, zero coefficient of transverse friction, we propose a triangular direct wave that is optimal. Between these two extremes, a variety of complex, locally optimal motions are found. Some of these can be classified as standing waves (or ratcheting motions).