Whenever there is a dissipative, or more generally, a non-conservative force, one that can't be derived from a potential in the form $$Q_{\text{pot}}=\frac{\mathrm{d}}{\mathrm{d}t}\Big(\frac{\partial U}{\partial \dot{q}} \Big)-\frac{\partial U}{\partial q},$$ we can generalize Hamilton's equations to include dissipative forces this way: $$\frac{\mathrm{d}q}{\mathrm{d}t}=\frac{\partial H}{\partial p}, \qquad \frac{\mathrm{d}p}{\mathrm{d}t}=-\frac{\partial H}{\partial q}+Q_{\mathrm{np}}. \tag1$$ I am now wondering what kind of geometrical structure would these equations generate? The geometric structure no longer looks so nice, since one can show that the Hamiltonian alone does no longer generate the time evolution. For a general $f(q,p,t)$ we now have $$\frac{\mathrm{d}f}{\mathrm{d}t}=\{f,H\}+Q_{\mathrm{np}}\frac{\partial f}{\partial p}+\frac{\partial f}{\partial t}, \tag2$$ as can be seen by using the chain rule. Equation (2) implies we cannot expect Liouville's theorem will hold in general. Since (1) is non-variational, I don't have any idea how could the theory of canonical transformations be developed, since it begins from demanding the Hamiltonian in the new variables has to satisfy the same variational principle as the one with the old variables.
TL;DR What if we took equations (1) as a starting point and began analyzing the geometric structure they generate?
