I'm looking for a derivation of the method of Lagrange multipliers as used in the calculus of variations for extremizing a functional subject to constraints. More specifically, I'm trying to understand the relationship between the "method of Lagrange multipliers" from standard calculus and the "Lagrange multipliers" introduced when solving for the generalized constraint forces of a system in Lagrangian dynamics. The method as applied to standard functions has a nice, intuitive explanation: at the extremum point, the gradient of the function to be extremized must be equal to some multiple of the gradient of the constraint function. Is there a similar explanation in the case of functionals (specifically, as applied to the action functional)? I don't have access to a real text on the calculus of variations, and every online resource I've found (this page, for example) simply states the method without providing a derivation or explanation.