1.2: Ecuaciones Diferenciales Ordinarias

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Set

$$E (v) =\ int_a^bf (x, v (x), v' (x))\ dx\]

y para dado$u_a,\ u_b\in{\mathbb R}$

$$V=\ {v\ en C^2 [a, b]:\ v (a) =u_a,\ v (b) =u_b\},\]

donde$y$ y$f$ sea suficientemente regular. Uno de los problemas básicos en el cálculo de la variación es

(P)$\min_{v\in V}E(v)$.

Ecuación de Euler

Dejar$u\in V$ ser una solución de (P), entonces

$$\ frac {d} {dx} f_ {u'} (x, u (x), u' (x)) =f_u (x, u (x), u' (x))\]

pulg$(a,b)$.

Ejercicio$\PageIndex{1}$: Proof

Para fijo$\phi\in C^2[a,b]$ con$\phi(a)=\phi(b)=0$ y real$\epsilon$,$|\epsilon|<\epsilon_0$, conjunto$g(\epsilon)=E(u+\epsilon \phi)$. Ya$g(0)\le g(\epsilon)$ que sigue$g'(0)=0$. La integración por partes en la fórmula para$g'(0)$ y el siguiente lema básico en el cálculo de variaciones implica la ecuación de Euler.

$Variaciones admisibles$

Figura 1.2.1.1: Variaciones admisibles

Lema básico en el cálculo de las variaciones. Let$h\in C(a,b)$ y

$$\ int_a^bh (x)\ phi (x)\ dx=0$$

para todos$\phi\in C_0^1(a,b)$. Entonces$h(x)\equiv0$ adelante$(a,b)$.

Comprobante. Asumir$h(x_0)>0$ por una$x_0\in (a,b)$, entonces hay$\delta>0$ tal que$(x_0-\delta,x_0+\delta)\subset(a,b)$ y$h(x)\ge h(x_0)/2$ en$(x_0-\delta,x_0+\delta)$.
Set

$$
\ phi (x)
=\ left\ {\ begin {array} {r@ {\ quad\ mbox {if}\ quad} l}
\ left (\ delta^2-|x-x_0|^2\ derecha) ^2 & x\ in (x_0-\ delta, x_0+\ delta)\\
0 & x\ in (a, b)\ setmenos [x_0-\ delta, x_0+\ delta]
\ end {array}\ right.
\]

Así$\phi\in C_0^1(a,b)$ y

$$\ int_a^b h (x)\ phi (x)\ dx\ ge\ frac {h (x_0)} {2}\ int_ {x_0-\ delta} ^ {x_0+\ delta}\ phi (x)\ dx>0,\]

lo que es una contradicción con el supuesto del lema.

$\Box$

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