In this Phys.SE question, one answer (by Ron Maimon) claims that one can make the assumption of a least action principle plausible using Liouville's Theorem as another starting point of the theory. The answer claims that conservation of information is equivalent to conservation of phase space volume (which is understandable and plausible to me), and follows from that that the time evolution of a system is given by a canonical transformation, and hence, by the canonical equations, in Hamilton formalism. I understand this part.
The answer then tries to provide an analog argument in the Lagrangian formalism, which I don't understand. It considers the space of all solutions in the configuration space as the phase space, but I quite don't understand the argument. Can anyone tell me how one can make plausible the derivation of the principle of least action, using the conservation of phase space volume (or something equivalent) in the Lagrangian formalism?
Edit: It would be nice if a possible answer could (as much as possible) make use of mathematics that are usually known to students of classical mechanics.