Question

Find the extreme functions, y(x),z(x) for the functional F[y,z]= integrate from 0 to 1 [y^2+z^2+4(y')^2-16(z')^2] dx

Answer 1

\[\triangle\]

Answer 2

i don't understand

Answer 3

hi @ghdoru11

that silly sign means I am watching and interested

i think you you want extreme functions y(x),z(x) for:

\[F[y,z]= \int_{0}^{1} y^2+z^2+4(y')^2-16(z')^2 \ dx\]

i could have a go but i think that might be really counter productive as i'd be guessing the relationship between x,y,z etc ie i don't recognise some of the finer lingo

so i can, in the first instance, tag some people who know way better than i and some way you will get some help

if that doesn't work out....

@Phi
@SithsAndGiggles

Answer 4

thank you for your time,i didn't know what that sign means

Answer 5

@Michele_Laino

Answer 6

hint:
here you have to write the Euler-Lagrange equation, for both functions y, and z, namely:
\[\Large \begin{gathered}
\frac{{\partial f}}{{\partial y}} - \frac{d}{{dx}}\frac{{\partial f}}{{\partial y'}} = 0 \hfill \\
\hfill \\
\frac{{\partial f}}{{\partial z}} - \frac{d}{{dx}}\frac{{\partial f}}{{\partial z'}} = 0 \hfill \\
\end{gathered} \]
where:
\[\Large f\left( {y,y'z,z'} \right) = {y^2} + {z^2} + 4{(y')^2} - 16{(z')^2}\]
furthermore, we need to know the initial conditions for \( \large y(x), \; z(x) \) at \( \large x=0, x=1 \)

Answer 7

please solve both the Euler-Lagrange differential equations, for \( \large y(x), \; z(x) \) respectively

Answer 8

I'm not well-versed in functional calculus, but it sounds really interesting...

Answer 9

this is wrong. right?