  Jamming is an emergent phenomenon wherein the local stability of individual
particles percolates to form a globally rigid structure. However, the onset of
rigidity does not imply that every particle becomes rigid, and indeed some
remain locally unstable. These particles, if they become unmoored from their
neighbors, are called \textit{rattlers}, and their identification is critical
to understanding the rigid backbone of a packing, as these particles cannot
bear stress. The accurate identification of rattlers, however, can be a
time-consuming process, and the currently accepted method lacks a simple
geometric interpretation. In this manuscript, we propose two simpler
classifications of rattlers based on the convex hull of contacting neighbors
and the maximum inscribed sphere of the radical Voronoi cell, each of which
provides geometric insight into the source of their instability. Furthermore,
the convex hull formulation can be generalized to explore stability in
hyperstatic soft sphere packings, spring networks, non-spherical packings, and
mean-field non-central-force potentials.
