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\begin{document}

Nutná podmínka pro autonomní úlohu:
je-li $f(x,y,z)=g(y,z)$ a $y$ stacionárním bodem $F$, potom existuje $C\in\er$, že
$$
g(y,y')-y'g_z(y,y')=C.
$$

\begin{thm}[o Lagrangeových multiplikátorech]
\Necht $f,g\in C^2([a,b]\times\er^2)$, $C\in\er$ a $y\in M$ je bodem minima funkcionálu $F$ vzhledem k množině
$$
\{y\in M: G(y)=C\},
$$
kde
$$
G(y)=\intab g(x,y(x),y'(x))\,dx,
$$
(zde analogicky k $\Phi$ značíme $\Psi(u)=G(u+\varphi)$).
Pokud $d\Phi(y)\not=0$, potom existuje $\lambda\in\er$, že
$$
d\Phi(y)(h)=\lambda d\Psi(y)(h),\quad h\in X\setminus\{0\}.
$$
\end{thm}

Dále se podívali na nějaké aplikace variačního počtu ve fyzice: úlohu o brachystochroně a úlohu o zavěšeném řetězu.

\end{document}