Mass, Impetus and Force by Symplectic Realizations of the Static Group

We show by symplectic realizations of the one dimensional Static group $G$ that the maximal $G$-elementary system is a a massive particle under an invariant force $f$ participating in the linear momentum and an invariant impetus $I$ participating in the change of position. Moreover the system is characterized by four physical quantities : a mass $m$, a boost $u=\frac{I}{m}$, an acceleration $a=\frac{f}{m}$ and an internal energy $U$. The hamiltonian of the system is a sum of the potential energy $V=-fq$ and a contribution $pu$ of the impetus.