I'm currently self studying calculus of variations and stumbled upon something called a Lagrange multiplier, used in solving isoperimetric problems and the like. Not knowing what they are I backtrack and found this wikipedia article.
Ok, so I see how they are use full in combination with calculus of variation, however the problem described by wikipedia can be solved by substituting a parametric representation of the constraint into the function and find extrema using normal calculus.
What are the limitations of this method (the substitution one) and why are Lagrange multipliers used instead? What additional classes of problem can they be applied to apart from the mentioned ones?