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Qns on euler-lagrangian equation

by almo
Tags: equation, eulerlagrangian
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almo
#1
Nov30-12, 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 euler-lagrangian eq considered as tough for an undergrad?
i am seriously struggling with it...
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Vargo
#2
Nov30-12, 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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