How should a snake turn on ice: a case study of the asymptotic isoholonomic problem

It is a classical result that solutions to the isoperimetric problem, i.e., finding the planar curves with a fixed length that enclose the largest area, are circles. As a generalization, we study an asymptotic version of the dual isoholonomic problem in a Euclidean space with a co-dimension one distribution. We propose the concepts of asymptotic rank and efficiency, and compute these quantities as well as the efficiency-achieving curves in several special cases. In particular, an example of a snake moving on ice is worked out in detail to illustrate the results.