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Both analytic work and recent $N$-body simulations have shown that a wide variety of galaxy models are subject to bending instabilities. We summarize this work, emphasizing the physics of bending instabilities, and propose a simple criterion for stability. We show that long-wavelength modes of inhomogeneous disk-like systems are not always stabilized by gravity, in contrast to those of a uniform sheet, and suggest instead that the most important stabilizing mechanism for galaxies is the out-of-phase response of stars that encounter the bend with a frequency greater than their free vertical oscillation frequency $\kappa_z$. This mechanism successfully accounts for our $N$-body results and for a number of previous theoretical results. It further predicts stability at all wavelengths to bending modes in pressure-supported systems in which the ratio of orbital oscillation frequencies is less extreme than about 2:1, i.e.\ for which the isodensity contours are rounder than about 1:3, in agreement with the behavior of bending modes in $N$-body models and in uniform spheroids, as well as with the absence of elliptical galaxies flatter than about E7. (Oblate systems can be flatter if the velocities are azimuthally biased). Bending instabilities in both barred and axisymmetric disks frequently create thicker bulge-like objects in the disk centers.
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