Illustrating Simple Variation Problem

Visualising It

But.. What does all these mean? Can we visualise it?

Note that the range of the functional here is \mathbb R – hence, we can easily visualise it with GeoGebra!

First we need to find a family of functions which satisfies the boundary condition – f(x)=x^n is ideal (and contains the extrema of our interest for n=3).

Now, we can find F(x,f(x),f'(x)), putting F(x)=f'^2+12x f(x) – geogebra will do the needful. If needed, we can restrict the function F(x) to the region of integration by giving F(x)=IF(0<=x<=1, f'^2+12x f(x)).

Now, the functional is v[f(x)]=\int_0^1 F (x,f(x),f'(x))  dx – which can be given in GeoGebra as v=Integral(F,0,1)

Open the above GeoGebra link to see a sample. There are two controls – one drop down (there are two family of functions which satisfies boundary conditions – x^n and \sin((2n+1){\pi\over2}x) – and a slider titled n. Try varying the slider and function – see when functional v has the least value!

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