The Principle of Least Action for Fields Containing Higher Order   Derivatives
  When generalizing the principle of least action for fields containing higher order derivatives, in general, it is not possible not to take into account the surface integrated term since it gives direct contribution to the forms of the equations of motion,of the energy-momentum tensor and of the angular-momentum tensor. This result is applied to two examples. In two dimensional gravity, it is essential to supplement Dirichclet condition for the surface integrated terms to vanish. On the contrary, in Hamiltonian description of the open rigid string, the equation of motion is modified by surface integrated terms. Boundary conditions are classified according to forms of certain equations of motion. 
Nguyen Duc Minh