  A rational map with good reduction in the field $\mathbb{Q}\_p$ of $p$-adic
numbers defines a $1$-Lipschitz dynamical system on the projective line
$\mathbb{P}^1(\mathbb{Q}\_p)$ over $\mathbb{Q}\_p$. The dynamical structure of
such a system is completely described by a minimal decomposition. That is to
say, $\mathbb{P}^1(\mathbb{Q}\_p)$ is decomposed into three parts: finitely
many periodic orbits; finite or countably many minimal subsystems each
consisting of a finite union of balls; and the attracting basins of periodic
orbits and minimal subsystems. For any prime $p$, a criterion of minimality for
rational maps with good reduction is obtained. When $p=2$, a condition in terms
of the coefficients of the rational map is proved to be necessary for the map
being minimal and having good reduction, and sufficient for the map being
minimal and $1$-Lipschitz. It is also proved that a rational map having good
reduction of degree $2$, $3$ and $4$ can never be minimal on the whole space
$\mathbb{P}^1(\mathbb{Q}\_2)$.
