  This work originates from part of a final year undergraduate research project
on the Eisenhart lift for Hamiltonian systems. The Eisenhart lift is a
procedure to describe trajectories of a classical natural Hamiltonian system as
geodesics in an enlarged space. We point out that it can be easily obtained
from basic principles of Hamiltonian dynamics, and as such it represents a
useful didactical way to introduce graduate students to several modern concepts
of geometry applied to physics: curved spaces, both Riemannian and Lorentzian,
conformal transformations, geometrisation of interactions and extra dimensions,
geometrisation of dynamical symmetries. For all these concepts the Eisenhart
lift can be used as a theoretical tool that provides easily achievable
examples, with the added benefit of also being a topic of current research with
several applications, among which the study of dynamical systems and
non-relativistic holography.
