The Routh of the Attractor Mechanism

We investigate and clarify various aspects of the effective dynamics of Maxwell-Einstein-scalar theories in the background of static, spherically symmetric and asymptotically flat extremal black holes in four space-time dimensions. This rigorously places the one-dimensional effective radial dynamics governed by the Attractor Mechanism, through the critical points of the Ferrara-Gibbons-Kallosh effective black hole potential $V_{BH}$, into the Routhian formalism, a framework which is intermediate between the Lagrange and Hamilton ones, based on a partial Legendre transform, and especially relevant in presence of cyclic variables. We elucidate and analyze the interplay of a trio of effective functionals: the aforementioned $V_{BH}$, Sen's entropy functional $\mathcal{E}$, and the relevant effective Routhian functional $\mathcal{R}$. Through their critical values at the event horizon, such functionals determine the Bekenstein-Hawking and the Wald entropy of the extremal black hole.