  Evolution equations which describe the changes in a velocity field over time
have been classically studied within the Eulerian or Lagrangian frame of
reference. Classically, these frameworks are equivalent descriptions of the
same problem, and the equivalence can be demonstrated by constructing particle
paths. For hyperbolic conservation laws, we extend the equivalence between
these frameworks to weak solutions for a broad class of problems. Our main
contribution in this paper is that we develop a new framework to extend the
idea of a particle path to scalar equations and to systems in one dimension
which do not explicitly include velocity fields. For systems, we use Riemann
invariants as the tool to develop an analog to particle paths.
