  In this paper, we review a general technique for converting the standard
Lagrangian description of a classical system into a formulation that puts time
on an equal footing with the system's degrees of freedom. We show how the
resulting framework anticipates key features of special relativity, including
the signature of the Minkowski metric tensor and the special role played by
theories that are invariant under a generalized notion of Lorentz
transformations. We then use this technique to revisit a classification of
classical particle-types that mirrors Wigner's classification of quantum
particle-types in terms of irreducible representations of the Poincar\'e group,
including the cases of massive particles, massless particles, and tachyons.
Along the way, we see gauge invariance naturally emerge in the context of
classical massless particles with nonzero spin, as well as study the massless
limit of a massive particle and derive a classical-particle version of the
Higgs mechanism.
