Relation between Euler's Elasticae and Sub-Riemannian Geodesics on SE(2)
  In this note we describe a relation between Euler's elasticae and sub-Riemannian geodesics on SE(2). Analyzing the Hamiltonian system of Pontryagin maximum principle we show that these two curves coincide only in the case when they are segments of a straight line. 
Alexey Mashtakov, Andrei Ardentov, Yuri Sachkov
math.OC