So the integral of the lagrangian over time must be stationary according to hamiltons principle. One can show that this leads to the euler lagrange equations, one for each pair of coordinates (qi,qi'). But my book has now started on defining a generalized lagrangian where lagrangian multipliers are used to somehow extend the principle to holonomic constraint f(q1,...,qn) = 0. My question is: Did the lagrangian not already work for holonomic constraints, if you took the displacements of the qi's to be independent? I should think so, so why is it that they want to start with these multipliers - are they trying to extend the lagrangian to work for systems in which you can use arbitrary displacements of the coordinates qi?