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Christ, that's _awesome_. I wish I was taught the Legendre transform this way. Where did you run into this viewpoint? I'd love to read a textbook that explains

by bollu 7y ago

Christ, that's _awesome_. I wish I was taught the Legendre transform this way. Where did you run into this viewpoint? I'd love to read a textbook that explains physics this way. In fact, I to this day have not found a book that caters to my level of mathematical knowledge (undergrad pure math) on variational calculus. They're either _way_ too handwavy (like Taylor), or _way_ beyond reach (relying on heavy functional analysis, weak convergence, all that stuff). Is there a textbook/lecture series/what have you that explains the calculus of variations at a "non-handwavy" level, BUT does not tries to perform the calculations in the most general way possible. That is, pick a nice space (C^\infinity or some such) and then show me with full rigor how the calculus of variations can be constructed. I'd love some references :)

jules 7y ago
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.

by bollu 7y ago

jules 7y ago

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.