Normally lagrange multipliers are used in the following sense. Suppose we are given a function f(x,y.z..,) and the constraint g(x,y,z,...,) = c Define a lagrange function: L = f - λ(g-c) And find the partial derivatives with respect to all variables and λ. This gives you the extrema since for an extrema ∇f = λ∇g However, I find that in my mechanics book this is used differently. I have attached a picture of the place where lagrange multipliers are used. I don't see how they in this case are used to maximize a quantity (which should be L) under a constraint like the above. Can anyone show me how they are and which gradients are to be parallel?