This result alone would make us believe that the system is indeed dissipative since it is clear that the allowed classical states would collapse to the zero volume in time. However, if there would be a local transformation to a set of canonical variables, a volume preserved phase-space would emerge. This phase-space would obey the Darboux and the Liouville theorems. The condition for the existence of such transformation is given by \(\left\{ x,y\right\} =e^{-\dot{G}t}\), where \(\left\{\bullet,\bullet\right\}\) are the Poisson brackets with respect to the variables \(\left(q,p\right)\). This condition is indeed quite general. However, the only allowed transformation that leads to the two first equations of (\ref{eq:02a}) is given by \(x=q\), and \(y=e^{-\dot{G}t/2}p\) provided \(G\) is homogeneous of degree zero. Both sets of first-order equations are not generally compatible.