Show, using the Poisson bracket formalism, that the Laplace-Runge-Lenz vector

\[\mathbf{A} \equiv \mathbf{p} \times \mathbf{L}-\frac{m k \mathbf{r}}{r}\]

is a constant of the motion for the Kepler problem of a particle moving in the central inverse-square force field \(F=-k / r^{2}\). Here \(\mathbf{p}\) is the particle's momentum, and \(\mathbf{L}\) is its angular momentum. Write \(L^{k}=\varepsilon^{k l n} x^{l} p^{n}\) and you might need to use the identity \(\varepsilon^{i j k} \varepsilon^{i l m}=\delta^{j l} \delta^{k m}-\delta^{j m} \delta^{k l}\).