R. M. Kiehn "An Extension of Hamilton's Principle to Include Dissipative
Systems" J. Math Phys Vol 15 no 1 (1974) p.9
Abstract: The key idea of conservative Hamiltionian systems is the fact
that the closed line integral of action is an absolute invariant of the motion.
Dissipation effects may be included by considering those systems for which the
closed integral of action is a parameter-dependent, conformal invaraint of the
motion. An application of this idea to hydrodynamics is made and the conditions
required for the validity of the Liouville theorem with respect to conformal
Hamiltonian flows are examined.