The first term in~\Eq{(58.2)} \begin{align*} &= \tfrac{1}{2} \{\mu\alpha, \beta\}\, \delta\left(\sqrt{-g} · g^{\mu\nu} g^{\alpha\epsilon} \left(\frac{\dd g_{\nu\epsilon}}{\dd x_{\beta}} + \frac{\dd g_{\beta\epsilon}}{\dd x_{\nu}} - \frac{\dd g_{\beta\nu}}{\dd x_{\epsilon}}\right)\right) \\ &= \tfrac{1}{2} \{\mu\alpha, \beta\}\, \delta\left(\sqrt{-g} · g^{\mu\nu} g^{\alpha\epsilon}\, \frac{\dd g_{\nu\epsilon}}{\dd x_{\beta}}\right) \\ &= -\tfrac{1}{2} \{\mu\alpha, \beta\}\, \delta\left(\sqrt{-g} · \frac{\dd g^{\mu\alpha}}{\dd x_{\beta}}\right) \quad\text{by \Eq{(35.11)}} \\ &= -\tfrac{1}{2} \{\mu\nu, \alpha\}\, \delta\left(\sqrt{-g} · \frac{\dd g^{\mu\nu}}{\dd x_{\alpha}}\right). \Tag{(58.31)} \end{align*} The second term reduces to the same. The third term becomes by~\Eq{(35.4)} \[ -\{\mu\nu, \alpha\}\, \delta\left(g^{\mu\nu}\, \frac{\dd}{\dd x_{\alpha}} \sqrt{-g}\right). \Tag{(58.32)} \] In the fourth term we have \[ g^{\mu\nu}\, \sqrt{-g}\, \{\mu\nu, \alpha\} = -\frac{\dd}{\dd x_{\nu}}(g^{\alpha\nu} \sqrt{-g}), \] by~\Eq{(51.41)}, since the divergence of~$g^{\alpha\nu}$ vanishes. Hence with some alterations of dummy suffixes, the fourth term becomes \[ \{\nu\beta, \beta\}\, g_{\mu}^{\alpha}\, \delta\left(\frac{\dd}{\dd x_{\alpha}} (g^{\mu\nu}\, \sqrt{-g})\right). \Tag{(58.33)} \] Substituting these values in~\Eq{(58.2)}, we have \begin{multline*} \delta\mf{L} = \bigl[-\{\mu\nu, \alpha\} + g_{\mu}^{\alpha} \{\nu\beta, \beta\}\bigr]\, \delta\left(\frac{\dd}{\dd x_{\alpha}} (g^{\mu\nu} \sqrt{-g})\right) \\ - \bigl[\{\mu\alpha, \beta\} \{\nu\beta, \alpha\} - \{\mu\nu, \alpha\} \{\alpha\beta, \beta\}\bigr]\, \delta(g^{\mu\nu} \sqrt{-g}). \Tag{(58.4)} \end{multline*} We write \[ \mf{g}^{\mu\nu} = g^{\mu\nu} \sqrt{-g};\quad \mf{g}_{\alpha}^{\mu\nu} = \frac{\dd}{\dd x_{\alpha}} (g^{\mu\nu} \sqrt{-g}). \Tag{(58.45)} \] Then when $\mf{L}$~is expressed as a function of the~$\mf{g}^{\mu\nu}$ and~$\mf{g}_{\alpha}^{\mu\nu}$, \Eq{(58.4)}~gives \begin{align*} \frac{\dd\mf{L}}{\dd\mf{g}^{\mu\nu}} &= -\bigl[\{\mu\alpha, \beta\} \{\nu\beta, \alpha\} - \{\mu\nu, \alpha\} \{\alpha\beta, \beta\}\bigr], \Tag{(58.51)} \\ \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}} &= \Neg\bigl[-\{\mu\nu, \alpha\} + g_{\mu}^{\alpha} \{\nu\beta, \beta\}\bigr]. \Tag{(58.52)} \end{align*} Comparing with~\Eq{(37.2)} we have \[ G_{\mu\nu} = \frac{\dd}{\dd x_{\alpha}}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}} - \frac{\dd\mf{L}}{\dd\mf{g}^{\mu\nu}}. \Tag{(58.6)} \] This form resembles that of Lagrange's equations in dynamics. Regarding \index{Lagrange's equations}% $\mf{g}^{\mu\nu}$ as a coordinate~$q$, and $x_{\alpha}$~as a four-dimensional time~$t$, so that $\mf{g}_{\alpha}^{\mu\nu}$ is a velocity~$q'$, the gravitational equations $G_{\mu\nu} = 0$ correspond to the well-known form \[ \frac{d}{dt}\, \frac{\dd\mf{L}}{\dd q'} - \frac{\dd\mf{L}}{\dd q} = 0. \] %\PageSep{133}