  The Eisenhart lift establishes a fascinating connection between
non-relativistic and relativistic physics, providing a space-time geometric
understanding of non-relativistic Newtonian mechanics. What is still little
known, however, is the fact that there is a Hilbert space representation of
classical mechanics (also called Koopman-von Neumann mechanics) that attempts
to give classical mechanics the same mathematical structure that quantum
mechanics has. In this article, we geometrize the Koopman-von Newmann (KvN)
mechanics using the Eisenhart toolkit. We then use a geometric view of KvN
mechanics to find transformations that relate the harmonic oscillator, linear
potential, and free particle in the context of KvN mechanics.
