Integrable dynamics and geometric conservation laws of hyperelastic strips

We consider the energy-minimizing configuration of the Sadowsky-type functional fornarrow rectifying strips. We show that the functional is proportional to thep-Willmore functional usingclassical analysis techniques and the geometry of developable surfaces. We introduce hyperelasticstrips (or p-elastic strips) as rectifying strips whose base curves are the critical points of the Sadowsky-type functional and find the Euler-Lagrange equations for hyperelastic strips using a variationalapproach. We show a naturally expected relationship between the planar stationary points of theSadowsky-type functional and the hyperelastic curves. We derive two conservation vector fields, theinternal force and torque, using Euclidean motions and obtain the first and second conservation lawsfor hyperelastic strips.