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A question on functionals

  1. Sep 30, 2014 #1
    First of all, apologies if this isn't quite in the right section.

    I've been studying functionals, in particular pertaining to variational calculus. My query relates to defining a functional as an integral over some interval [itex]x\in [a,b][/itex] in the following manner [tex]I[y]= \int_{a}^{b} F\left(x, y(x), y'(x)\right)dx [/tex]
    Clearly from this we see that [itex]I[/itex] is not dependent on [itex]x[/itex], but instead it depends only on the function [itex]y(x)[/itex]. [itex]I[/itex] is a functional and as such it defines a mapping from the set of all functions [itex]y(x)[/itex] satisfying [itex]y(a)=0=y(b)[/itex] to [itex]\mathbb{R}[/itex].

    My question really, is why the integrand a function of the set of curves [itex]y(x)[/itex] (as defined above) and their derivatives [itex]y'(x)[/itex] (I've kept it to first-order for simplicity, but I know that in general it can be dependent on higher orders)?

    Is this because, as [itex]I[/itex] is depends on every single value that [itex]y(x)[/itex] takes in the interval [itex]x\in [a, b][/itex], and not just its value at a single point, we must consider how [itex]y(x)[/itex] changes (i.e. we must consider it's derivatives) over this interval as we integrate over it. Thus, this implies that the integrand should be a function of the curve and it's rate of change?

    Please could someone let me know if my thinking is correct, and if not, provide an explanation.

    Thanks for your time.
  2. jcsd
  3. Oct 1, 2014 #2
    Sorry, I meant the set of all functions [itex]y(x)[/itex] satisfying [itex]\delta y(a)=0=\delta y(b)[/itex] to [itex]\mathbb{R}[/itex].
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