Find a canonical drag/measure in contact dynamics obtained by removing dynamical similarity

Identify and justify a principled choice of contact form—equivalently, a choice of drag—that uniquely determines a natural, gauge-invariant measure on contact state space (arising from quotienting by dynamical similarity) with theoretical virtues analogous to the Liouville measure in symplectic mechanics, thereby minimizing conventionality in time-dependent typicality assessments.

Background

Quotienting by dynamical similarity transforms Hamiltonian systems into contact systems where the natural Liouville measure becomes time-dependent and non-unique, reflecting a freedom to choose the contact form (drag). This conventionality contrasts with the symplectic case, where the Liouville measure is uniquely singled out by invariance and conservation under all Hamiltonian flows.

The thesis demonstrates that different choices of drag can lead to families of time-dependent measures with desirable properties for explaining arrows of time, but emphasizes that no unique choice has been identified. Establishing a canonical drag would recover virtues of the symplectic Liouville measure—simplicity, universality, uniqueness—and strengthen explanatory claims.