Chetaev vs. vakonomic prescriptions in constrained field theories with   parametrized variational calculus
  Starting from a characterization of admissible Cheataev and vakonomic variations in a field theory with constraints we show how the so called parametrized variational calculus can help to derive the vakonomic and the non-holonomic field equations. We present an example in field theory where the non-holonomic method proved to be unphysical. 
Enrico Bibbona, Lorenzo Fatibene, Mauro Francaviglia