  The dynamics of amorphous granular matter with frictional interactions cannot
be derived in general from a Hamiltonian and therefore displays oscillatory
instabilities stemming from the onset of complex eigenvalues in the stability
matrix. These instabilities were discovered in the context of one and two
dimensional systems, while the three dimensional case was never studied in
detail. Here we fill this gap by deriving and demonstrating the presence of
oscillatory instabilities in a three dimensional granular packing. We study
binary assemblies of spheres of two sizes interacting via classical Hertz and
Mindlin force laws for the longitudinal and tangent interactions, respectively.
We formulate analytically the stability matrix in 3D and observe that a couple
of complex eigenvalues emerges at the onset of the instability as in the case
of frictional disks in two-dimensions. The dynamics then shows oscillatory
exponential growth in the Mean-Square-Displacement, followed by a catastrophic
event. The generality of these results for any choice of forces that break the
symplectic Hamiltonian symmetry is discussed.
