  This is the second paper devoted to energetic rigidity, in which we apply our
formalism to examples in two dimensions: underconstrained random regular spring
networks, vertex models, and jammed packings of soft particles. Spring networks
and vertex models are both highly underconstrained, and first-order constraint
counting does not predict their rigidity, but second-order rigidity does. In
contrast, spherical jammed packings are overconstrained and thus first-order
rigid, meaning that constraint counting is equivalent to energetic rigidity as
long as prestresses in the system are sufficiently small. Aspherical jammed
packings on the other hand have been shown to be jammed at hypostaticity, which
we use to argue for a modified constraint counting for systems that are
energetically rigid at quartic order.
