Why do so many galaxies in clusters have a near zero velocity? I'm looking at a velocity chart of the Coma Cluster:
 
And the question occurred to me: why are there so many galaxies that have a zero velocity (relative to the core of the Coma Cluster which is roughly 7000 km s-1.  How can they be in 'equilibrium'?  At that distance you would expect to find galaxies at the peak of their red or blue shift.  What am I missing?
 A: For this answer I'm going to pretend that the y-axis of the plot you've shown has had $\sim 7000\,{\rm km}\,{\rm s}^{-1}$ subtracted - this is a conventional thing to do in diagrams like the one you show, and simplifies the explanation/interpretation. 
The plot you show is based on observations. When observing distant objects like Coma, only 3 of the 6 phase space coordinates are available. One can measure two spatial coordinates in the plane of the sky (giving the $R$ on the plot), but the distance is inaccessible. And one can measure the velocity along the line of sight via the redshift, but the velocity in the plane of the sky ("proper motion") is completely inaccessible at the distance of Coma. And even the line of sight velocity is tricky: it is a combination of the comoving velocity (from Hubble's law) and the peculiar velocity (e.g. from motion within Coma). This means you might sometimes confuse a background galaxy (further away -> higher comoving velocity) with a Coma galaxy (closer -> lower comoving velocity, but large peculiar velocity).
The $v$ shown is the component of the velocity offset from the systemic velocity of Coma along the line of sight. So many of the galaxies with $v=0$ actually have large velocities, but directed perpendicular to the line of sight. This is especially true for galaxies with small $R$ (at large $R$ there is a higher probability that the galaxy is actually at apocentre and has a true zero velocity offset from the cluster core).
Some will argue that you can measure distances, but most distance measures that work at cluster distances are so uncertain that all you can tell is that the galaxy is probably at the distance of the cluster - not where it is in the cluster. E.g. you cannot know whether the galaxy is on the inward portion of its orbit and behind the cluster, or on the outward portion of its orbit and in front of the cluster - all you have is a velocity offset from the cluster centre.
If you want additional depth, figures, and so on you might try an extragalactic astronomy textbook, or if you're feeling adventurous my MSc thesis has a couple of section (3.2-3.3) dedicated to these projection issues.
A: Firstly, let's define what's on the axes (thanks to Kyle Oman for prompting me to do this):
The velocity on the $y$ axis is not the total velocity of a galaxy through space; it is only the component of the velocity along the line of sight (LOS), measured by the redshift of the galaxy. Let's call it $v_z$, although on your figure it's just called $v$. The components perpendicular to the LOS, $v_x$ and $v_y$, can only be measured if you have the patiency to wait for a galaxy to move enough across the sky. To move 1 arcsec across the sky (which is roughly what you need to see a difference) at 1000 km s$^{-1}$ will take half a million years.
This means that a galaxy may be plummeting through the cluster at tremendous $v_x$ and $v_y$, but you will measure $v_z$ to be $\sim0$ (relative to the systemic velocity of the cluster, which in this case is 6930 km s$^{-1}$).
On the $x$ axis we have $R$, the projected distance from the center of the cluster to a given galaxy, measured in arcmin.
Galaxies with low total (kinetic + potential) energy have low total velocities ($\sqrt{v_x^2+v_y^2+v_z^2}$), and thus also low $v_z$. They hang out close to the center of the cluster, since they don't have the energy to achieve large distances. Thus they appear to the left in the plot, at middle height.
Galaxies with high total energy may either orbit slowly at large distances (i.e. they're found to the right in the plot, but at middle height), or, if they have more radial trajectories, they have low velocities at large distances but then speed up as they approach the center. If their trajectories are mostly along the LOS, they're found to the left in the plot. If their trajectories are mostly perpendicular to the LOS, they're found both left, middle, and right (all $R$-values), but always at middle height (low $v_z$-values). After they pass the center, they lose kinetic energy as they again increase their distance.
That is, the largest blue- and redshifts are found for galaxies that have high energies at radial trajectories along the LOS, and which are currently close to the center of the cluster, i.e. at low $R$.
