I don't think I've seen this viewpoint anywhere else, but Langrange multipliers in the calculus of variations are a well known thing, and the connection between

I don't think I've seen this viewpoint anywhere else, but Langrange multipliers in the calculus of variations are a well known thing, and the connection between Lagrange multipliers and the Legendre transform is a well known thing, so I'm sure you can find it in a book somewhere. I've not read any book on the calculus of variations, but I think that the book by Gelfand and Fomin is the standard book. Maybe that is suitable? By the way, you can motivate the Legendre transform / Langrange multipliers like this. Suppose we want to minimise f(x) subject to g(x) = 0. We can add the constraint g(x) = 0 to the objective function by defining Q(x) = 0 if g(x) = 0 and Q(x) = infinite otherwise. Then the problem is min f(x) + Q(x). This seems like the kind of thing that doesn't help, but now write Q(x) = max_p (p g(x)). You can see that this is indeed the same as the previous definition of Q. So we can write that problem as min_x max_p f(x) + p g(x). This is a bit silly, and the usual way to explain Lagrange multipliers is probably better, but you may find it amusing.