Chapter 4.P4, Problem 6P

Carry out the calculations that lead from Eq. $\left(8\right)$ to Eq. $\left(11\right)$ in the discussion preceding Problem $4$.

$\begin{array}{l}\varphi \left(\epsilon \right)=J\left(\bar{x}+\epsilon \xi ,\bar{y}+\epsilon \eta \right)\\ \begin{array}{cc}& \end{array}={\displaystyle \underset{a}{\overset{b}{\int }}F\left(t,\bar{x}+\epsilon \xi ,\bar{y}+\epsilon \eta ,{\bar{x}}^{\prime }+\epsilon {\xi }^{\prime },{\bar{y}}^{\prime }+\epsilon {\eta }^{\prime }\right)},\end{array}$ $\left(8\right)$

where the perturbation vector $\langle \xi \left(t\right),\eta \left(t\right)\rangle$ satisfies

$\langle \xi \left(a\right),\eta \left(a\right)\rangle =\langle 0,0\rangle$ and $\langle \xi \left(b\right),\eta \left(b\right)\rangle =\langle 0,0\rangle .$ $\left(9\right)$

Thus, for $\varphi \left(\epsilon \right)$ defined by Eq. $\left(8\right)$, we compute ${\varphi }^{\prime }\left(0\right)=0$, integrate by parts, and use the endpoint conditions $\left(9\right)$ to arrive at

${\displaystyle \underset{a}{\overset{b}{\int }}\left\{\left[F_x-F_{x^{\prime }t}\right]\xi +\left[F_y-F_{y^{\prime }t}\right]\eta \right\}dt=0.}$ $\left(10\right)$

Since Eq. $\left(10\right)$ must hold for any $\xi$ and $\eta$ satisfying the conditions $\left(9\right)$, we conclude that each factor multiplying $\xi$ and $\eta$ in the integrand must equal zero,

$\begin{array}{c}{F}_x-F_{x^{\prime }t}=0\\ F_y-F_{y^{\prime }t}=0.\end{array}$ $\left(11\right)$