
#1
Nov3012, 09:57 AM

P: 8

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 eulerlagrangian eq considered as tough for an undergrad? i am seriously struggling with it... 



#2
Nov3012, 11:10 AM

P: 350

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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