(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 \(I=\int_{x_{1}}^{x_{2}} F d x\) 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 \(2[Y=y+\epsilon \eta(x) \text { with } \eta(x)
\text { arbitrary }]\) and construct a similar formula for \(z\) llet
\(Z=z+\epsilon \zeta(x),\) where \(\zeta(x)\) is another arbitrary function].
Carry through the details of differentiating with respect to \(\epsilon,\)
putting \(\epsilon=0,\) 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
$$\int_{y_{1}}^{y_{2}} \int_{x_{1}}^{x_{2}} F\left(u, x, y, u_{x},
u_{y}\right) d x d y$$.
Hint: Let the varied \(U(x, y)=u(x, y)+\epsilon \eta(x, y)\) 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 \(x=y=y=y\), with respect to \(\epsilon,\) set
\(\epsilon=0,\) 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
$$\frac{d^{2}}{d x^{2}} \frac{\partial F}{\partial y^{\prime
\prime}}-\frac{d}{d x} \frac{\partial F}{\partial y^{\prime}}+\frac{\partial
F}{\partial y}=0$$
