Question on Calculus of variations formalism

  • #1

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Hi all, I had a quick question regarding the formalism behind calculus of variations. In one-dimensional standard calc, we consider functions [tex] f:\mathbb{R}\to \mathbb{R} [/tex] and define their derivatives using the conventional definition with the limit of the quotient of the change in the function, and the change of the function's parameter.

When we move on to calculus of variations, we instead consider functions (which we for some reason call functionals) [tex] F:\mathcal{F}(\mathbb{R})\to\mathbb{R} [/tex] where [itex] \mathcal{F}(\mathbb{R}) [/itex] is the set of all functions that map [itex] \mathbb{R}\to\mathbb{R} [/itex]. We define the derivative of these functions in an analogous way to how we did for real one-dimensional real functions. (And usually we'll be interested in finding stationary points, but that's another story.) Is this basically it?

I'm posting this because 1) I was confused because in physics textbooks, I recall reading things like "calculus of variations is about minimizing functionals, which are different from functions". But I'm trying to check to see if what they actually meant is that functionals are different from real functions (since it seems like functionals are just another specific type of function). And 2) because initially calculus of variations seemed like an especially hard thing, postponed for later in physics education when one was ready for things like Lagrangian mechanics; but it's starting to look like just another version of calculus, using a different set of functions to study! So I just wanted to check that if it is this straightforward :)
 
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  • #2
andrewkirk
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You are correct that a functional is just a particular class of function. I wouldn't go so far as to say that Calculus of Variations is 'just another version of calculus' though. There are important analogous elements but there are major differences as well.
 
  • #3
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When I was in college, a half a century ago, the word "functional" never came up. Now, it's standard fare. The thing is that the "universe of discourse" of differential calculus is NOT the real numbers; it is the space of all functions (continuous, well-behaved, smooth, etc.). So, logically you move from considering maps "from numbers to numbers" to maps "from functions to functions".
Usually CoV is presented after differential calculus of 1 variable, multvariate differentiation, integral calculus of 1 variable, and ordinary differential equations. Generally, partial differential equations (multivariate by definition) is taught along with or prior to CoV because they go hand in hand.
Anyway, I learned CoV without any need for the concept of "functional" although once I heard about it I saw how it allowed a more rigorous development of the subject. So, you're not wrong.
 

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