  Amorphous solids tend to present an abundance of soft elastic modes, which
diminish their transport properties, generate heterogeneities in their elastic
response, and affect non-linear processes like thermal activation of
plasticity. This is especially true in packings of particles near their jamming
transition, for which effective medium theory and variational arguments can
both predict the density of vibrational modes. However, recent numerics support
that one hypothesis of the variational argument does not hold. We provide a
novel variational argument which overcomes this problem, and correctly predicts
the scaling properties of soft modes near the jamming transition. Soft modes
are shown to be related to the response to a local strain in more connected
networks, and to be characterized by a volume $1/\delta z$, where $\delta z$ is
the excess coordination above the Maxwell threshold. These predictions are
verified numerically.
