  We numerically investigate the mechanical properties of static packings of
ellipsoidal particles in 2D and 3D over a range of aspect ratio and compression
$\Delta \phi$. While amorphous packings of spherical particles at jamming onset
($\Delta \phi=0$) are isostatic and possess the minimum contact number $z_{\rm
iso}$ required for them to be collectively jammed, amorphous packings of
ellipsoidal particles generally possess fewer contacts than expected for
collective jamming ($z < z_{\rm iso}$) from naive counting arguments, which
assume that all contacts give rise to linearly independent constraints on
interparticle separations. To understand this behavior, we decompose the
dynamical matrix $M=H-S$ for static packings of ellipsoidal particles into two
important components: the stiffness $H$ and stress $S$ matrices. We find that
the stiffness matrix possesses $N(z_{\rm iso} - z)$ eigenmodes ${\hat e}_0$
with zero eigenvalues even at finite compression, where $N$ is the number of
particles. In addition, these modes ${\hat e}_0$ are nearly eigenvectors of the
dynamical matrix with eigenvalues that scale as $\Delta \phi$, and thus finite
compression stabilizes packings of ellipsoidal particles. At jamming onset, the
harmonic response of static packings of ellipsoidal particles vanishes, and the
total potential energy scales as $\delta^4$ for perturbations by amplitude
$\delta$ along these `quartic' modes, ${\hat e}_0$. These findings illustrate
the significant differences between static packings of spherical and
ellipsoidal particles.
