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Qns on euler-lagrangian equation

  1. Nov 30, 2012 #1
    I find it hard to undestand the various notation used for the equation.
    Am i wrong to understand the equation as finding the maxima or the minima of an function?
    However, the terms like functional and small real parameter confuses me.
    I read up on whats a functional and cant really understand, so far my understanding of its, is that its a function where by instead of x, a varible, it consist of vectors like velocity and etc. Thus, am i wrong to say equation of KE is actually a functional?
    On the part of small real parameter ε.. i just have no idea. All i can infer is that is a change in the vector. But where is there this need to implictly express such a term?
    Is euler-lagrangian eq considered as tough for an undergrad?
    i am seriously struggling with it...
     
  2. jcsd
  3. Nov 30, 2012 #2
    Do you know what a vector space is? A functional is a map whose domain is a subset of a vector space and which takes scalar values.

    In the context of your question a typical vector space would be the set of differentiable functions on the interval [0,1].
    [itex] V =\{ y(x)| y\, \text{is differentiable in a neighborhood of the interval}\, [0,1]\}[/itex]

    An example of a functional would be a map [itex] \mathcal{F}(y)[/itex] with domain
    [itex] \{y\in V| y(0)=1,\, y(1)=5\}[/itex] and which is defined by a formula such as
    [itex] \mathcal{F}(y) = \int_a^b \sqrt{1+(y')^2}\, dx [/itex]

    In plainer language, in the context of calculus of variations, functionals take ordinary functions as inputs and return numbers as outputs.

    A good basic reference would be Gelfand "Calculus of Variations".
     
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