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 $v=Integral(F,0,1)$

View File in GeoGebra

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!