An Inverse Problem of the Calculus of Variations on Arbitrary Time   Scales
  We consider an inverse extremal problem for variational functionals on arbitrary time scales. Using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variational functional that attains a local minimum at a given point of the vector space. 
Monika Dryl, Agnieszka B. Malinowska, Delfim F. M. Torres
math.OC