Chapter 9.5, Problem 1P

(a) Consider the case of two dependent variables. Show that if $F=F\left(x,y,z,y^{\prime },z^{\prime }\right)$ and we want to find $y(x)$ and $z(x)$ to make $I={\displaystyle {\int }_{x_1}^{x_2}F}dx$ stationary, then $y$ and $z$ should each satisfy an Euler equation as in (5.1). Hint: Construct a formula for a varied path $Y$ for $y$ as in Section 2 $[Y=y+e\eta (x)\text{ with }\eta (x)$ arbitrary] and construct a similar formula for $z$ [let $Z=z+ϵ\zeta (x),$ where $\zeta (x)$ is another arbitrary function]. Carry through the details of differentiating with respect to $ϵ,$ putting $ϵ=0,$ and integrating by parts as in Section 2 ; then use the fact that both $\eta (x)$ and $\zeta (x)$ are arbitrary to get (5.1).

(b) Consider the case of two independent variables. You want to find the function $u(x,y)$ which makes stationary the double integral ${\displaystyle {\int }_{y_1}^{y_2}{\displaystyle {\int }_{x_1}^{x_2}F}}\left(u,x,y,u_x,u_y\right)dxdy$.

Hint: Let the varied $U(x,y)=u(x,y)+e\eta (x,y)$ where $\eta (x,y)=0$ at $x=x_1$ $x=x_2,y=y_1,y=y_2,$ but is otherwise arbitrary. As in Section 2, differentiate with respect to $ϵ$, set $ϵ=0$, integrate by parts, and use the fact that $\eta$ is arbitrary. Show that the Euler equation is then $\frac{\partial }{\partial x}\frac{\partial F}{\partial u_x}+\frac{\partial }{\partial y}\frac{\partial F}{\partial u_y}-\frac{\partial F}{\partial u}=0$.

(c) Consider the case in which F depends on $x,y,y^{\prime },$ and $y^{\prime \prime }.$ Assuming zero values of the variation $\eta (x)$ and its derivative at the endpoints $x_1$ and $x_2$, show that then the Euler equation becomes

$\frac{d^2}{d{x}^2}\frac{\partial F}{\partial y^{\prime \prime }}-\frac{d}{dx}\frac{\partial F}{\partial y^{\prime }}+\frac{\partial F}{\partial y}=0$.