ar X iv : s ol v - in t / 9 40 70 05 v 1 2 7 Ju l 1 99 4 Integrable dynamics of a discrete curve and

We show that the following elementary geometric properties of the motion of a discrete (i.e. piecewise linear) curve select the integrable dynamics of the Ablowitz-Ladik hierarchy of evolution equations: i) the set of points describing the discrete curve lie in the sphere S; ii) the distance between any two subsequent points does not vary in time; iii) the dynamics does not depend explicitly on the radius of the sphere. These results generalize to a discrete context our previous work on continuous curves [1]. ∗This work was supported by the 1993 agreement between Rome and Warsaw Universities; by the grant 2-0168-91-01 KBN and by the INFN.