  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 find that retrograde
traveling waves are optimal. The optimal snake deflection scales as the -1/4
power of the coefficient of transverse friction, in agreement with an
asymptotic analysis. At the other extreme, zero coefficient of transverse
friction, we propose a triangular direct wave which 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).
