  We prove some detailed quantitative stability results for the contact set and
the solution of the classical obstacle problem in $\mathbb{R}^n$ ($n \ge 2$)
under perturbations of the obstacle function, which is also equivalent to
studying the variation of the equilibrium measure in classical potential theory
under a perturbation of the external field.
  To do so, working in the setting of the whole space, we examine the evolution
of the free boundary $\Gamma^t$ corresponding to the boundary of the contact
set for a family of obstacle functions $h^t$. Assuming that $h=h^t (x) =
h(t,x)$ is $C^{k+1,\alpha}$ in $[-1,1]\times \mathbb{R}^n$ and that the initial
free boundary $\Gamma^0$ is regular, we prove that $\Gamma^t$ is twice
differentiable in $t$ in a small neighborhood of $t=0$. Moreover, we show that
the "normal velocity" and the "normal acceleration" of $\Gamma^t$ are
respectively $C^{k-1,\alpha}$ and $C^{k-2,\alpha}$ scalar fields on $\Gamma^t$.
This is accomplished by deriving equations for these velocity and acceleration
and studying the regularity of their solutions via single and double layers
estimates from potential theory.
