Differential equation linked to a functional – Natural functional of Euler-Lagrange.

Given a functional depending on a function

*f* =

*f*(

*x*), on its derivative

*f* _{ x } =

*df*/

*dx*, on the own variable

*x* and whose mathematical expression has the following form,

$$ \uppi ={\displaystyle \underset{x=a}{\overset{x=b}{\int }}I\left(f,\;{f}_x,\;x\right)}\kern0.24em dx $$

(4.67)

and making stationary its first variation,

$$ \updelta \uppi =0={\displaystyle \underset{x=a}{\overset{x=b}{\int }}\left[\frac{\partial I}{\partial f}\kern0.24em \updelta f+\frac{\partial I}{\partial {f}_x}\kern0.24em \updelta {f}_x\right]}\; dx $$

(4.68)

after integrating by parts it is obtained

\( {\displaystyle \underset{x=a}{\overset{x=b}{\int }}\frac{\partial I}{\partial {f}_x}\updelta {f}_x\; dx}={\left.\frac{\partial I}{\partial {f}_x}\;\updelta {f}_x\right|}_{x=a}^{x=b}-{\displaystyle \underset{x=a}{\overset{x=b}{\int }}\frac{d}{dx}\left(\frac{\partial I}{\partial {f}_x}\right)\kern0.24em \updelta f\; dx}, \), the following expression for the first variation,

$$ \updelta \uppi =0={\displaystyle \underset{x=a}{\overset{x=b}{\int }}\left[\frac{\partial I}{\partial f}-\frac{d}{dx}\left(\frac{\partial I}{\partial {f}_x}\right)\right]}\ \updelta f\; dx+{\left.\frac{\partial I}{\partial {f}_x}\;\updelta {f}_x\right|}_{x=a}^{x=b} $$

(4.69)

As δ

*f* ≠ 0, then the following relations are obtained,

$$ \begin{array}{l}\mathrm{a}\Big)\kern0.5em \mathrm{The}\ \mathrm{Euler}\hbox{-} \mathrm{Lagrange}\ \mathrm{equation}\kern0.75em \frac{\partial I}{\partial f}-\frac{d}{dx}\left(\frac{\partial I}{\partial {f}_x}\right)=0\kern1em ,\forall a\le x\le b,\\ {}{\left.\mathrm{b}\Big)\kern0.5em \mathrm{The}\ \mathrm{boundary}\ \mathrm{condition}\kern2.75em \frac{\partial I}{\partial {f}_x}\kern0.24em \updelta {f}_x\right|}_{x=a}^{x=b}=0\kern1.25em \mathrm{in}:\ x=a\kern1em \mathrm{y}\kern0.75em x=b\end{array} $$

(4.70)