The invariant integral \[ A = \tfrac{1}{4} \int F^{\mu\nu} F_{\mu\nu} \sqrt{-g}\, d\tau \Tag{(79.1)} \] is called the action of the electromagnetic field. In Galilean coordinates it becomes by~\Eq{(77.3)} \[ \int dt \iiint \tfrac{1}{2}(\alpha^{2} + \beta^{2} + \gamma^{2} - X^{2} - Y^{2} - Z^{2})\, dx\, dy\, dz. \Tag{(79.2)} \] Regarding the magnetic energy as kinetic~($T$) and the electric energy as potential~($V$) this is of the form \[ \int (T - V)\, dt, \] i.e.\ the time-integral of the Lagrangian function\footnotemark.\footnotetext {In dynamics there are two integrals which have the stationary property under proper restrictions, viz.\ $\int T\, dt$ and $\int (T - V)\, dt$. The first of these is the action as originally defined. In the general theory the term has been applied to both integrals somewhat indiscriminately, since there is no clear indication of energy which must be reckoned as potential.} The derivation of the electromagnetic equations by the stationary variation of this integral has been investigated in the classical researches of Larmor\footnotemark.\footnotetext {\Title{Aether and Matter}, Chapter~\Vol{VI}.} We shall now show that the two most important electromagnetic tensors, viz.\ the energy-tensor~$E^{\mu\nu}$ and the charge-and-current vector~$J^{\mu}$, are the Hamiltonian derivatives of the action, the formulae being \begin{align*} \frac{\Ham}{\Ham g_{\mu\nu}} (\tfrac{1}{4}F^{\mu\nu} F_{\mu\nu}) &= \tfrac{1}{2} E^{\mu\nu}, \Tag{(79.31)} \\ \frac{\Ham}{\Ham \kappa_{\mu}} (\tfrac{1}{4}F^{\mu\nu} F_{\mu\nu}) &= -J^{\mu}. \Tag{(79.32)} \end{align*} \PageSep{188} First consider small variations~$\delta g_{\mu\nu}$, the~$\kappa_{\mu}$ remaining constant. The~$F_{\mu\nu}$ (but not the~$F^{\mu\nu}$) will accordingly remain unvaried. We have then \begin{multline*} \delta (F^{\mu\nu} F_{\mu\nu} \sqrt{-g}) \\ \begin{aligned} &= F^{\mu\nu} F_{\mu\nu}\, \delta(\sqrt{-g}) + F_{\alpha\beta} F_{\mu\nu} \sqrt{-g} · \delta(g^{\mu\alpha} g^{\nu\beta}) \\ % &= F^{\sigma\tau} F_{\sigma\tau} \sqrt{-g} · \frac{1}{2}\, \frac{\delta g}{g} + F_{\alpha\beta} F_{\mu\nu} \sqrt{-g} (g^{\mu\alpha}\, \delta g^{\nu\beta} + g^{\nu\beta}\, \delta g^{\mu\alpha}) \\ &= \sqrt{-g} \{-\tfrac{1}{2} F^{\sigma\tau} F_{\sigma\tau} g_{\nu\beta}\, \delta g^{\nu\beta} + 2F_{\alpha\beta} F_{\mu\nu} g^{\mu\alpha}\, \delta g^{\nu\beta}\} \\ &= 2\sqrt{-g} · \delta g^{\nu\beta} \{-\tfrac{1}{4} g_{\nu\beta} F^{\sigma\tau} F_{\sigma\tau} + {F^{\mu}}_{\beta} F_{\mu\nu}\} \\ &= -2E_{\nu\beta} \sqrt{-g} · \delta g^{\nu\beta} \qquad\text{by~\Eq{(77.2)}} \\ &= \Neg2E^{\nu\beta} \sqrt{-g} · \delta g_{\nu\beta} \qquad\text{by~\Eq{(35.2)}.} \end{aligned} \end{multline*} From this \Eq{(79.31)}~follows immediately.