Magnetic levitation for a dipole
In the core of any passive magnetic levitation system using permanent magnets who magnetizations are fixed, it is a constant magnetic dipole 
and it couples to the ambient magnetic fields, which provides the desired levitation force. The dipole-fields interaction energy is given by 
. The corresponding force is simply the derivative of the interaction energy with respect to the dipole position, namely 
. One might expect that the magnetic dipole can be levitated as long as the DC gravitation force is balanced by the dipole-field force. However, such levitation is unstable in the free space, for 
. A stable levitation requires that the force lines merge into some point in the space, namely divergence of the force to be smaller than zero (
). This is the famous Earnshaw's theorem http://en.wikipedia.org/wiki/Earnshaw%27s_theorem.
Levitation with feedback control
In order to create a stable levitation, we need to apply a feedback control, which is more closely related to the realistic suspension scheme. In this case, the field is time-dependent and the Earnshaw's theorem is no longer relevant. Three related papers about the feedback control scheme are attached. The second and third are less technical, and a simple control scheme was proposed and demonstrated.
Trumper_linearization_suspension_control_1997.pdf
Guy_Marsdon_feedback_suspension_2003.pdf
Lilienkamp_feedback_mag_suspension_2004.pdf
In the figure below, it is the schematic setup in the third paper.
Qualitative Understanding of the Proposed Scheme
In the proposed scheme, we will simply use commercially-available permanent magnets which can be find in http://www.kjmagnetics.com/categories.asp. A stable suspension is achieved by feedback control as mentioned.
There are many possible configurations for a magnetic suspension. However, to achieve a low eigen-frequency, one must seek such a scheme that the magnetic force can achieve extrema at some spatial point and vary smoothly around the extrema when the position of the suspended magnets drift. After investigating many possible schemes, we found out that the simple scheme in which two cylinder magnets aligned in z-axis has such property. It is shown schematically in the following figure.
We can gain some qualitative understanding of such scheme by looking at the interaction between two current loops, which approximately represents the situation with two cylinder magnets. The corresponding loop-loop interaction is shown in the figure below.
There is actually an analytical formula for such interaction, which is given by
Here 
and 
are the complete elliptic integral of the first and second kind, respectively. The DC magnetic force as a function of the separation between two current loops are shown in the figure below. As we can see, if the suspended magnet is around the extrema, the rigidity at z-direction, or equivalently the eigen-frequency, is equal to 
. Due to the fact that 
, this also means that the rigidity at x- and y-direction are very small for 
due to approximate symmetry around z-axis.
Diamagnetism
An alternative way to achieve a stable suspension is to use diamagnetic materials, whose magnetic susceptibility is negative (
). The interaction energy is no longer the simple dipole-fields interaction. Ideally, the magnetization for those material is proportional to the magnetic field strength, namely 
. Therefore, the interaction energy is proportional to square of the magnetic field amplitude. The stable trapping requirement is simply 
, which indicates that 
(a minimum field amplitude). Therefore, as long as the magnetic field amplitude is minimum in some point and the DC magnetic force balances the gravitation force, the diamagnetic material will be stably suspended. For more details, one can look at http://en.wikipedia.org/wiki/Diamagnetic. As a distinguishing example of diamagnetism, the superconductor has 
and the superconductor can be easily stably levitated. One interesting superconducting levitated train can be found in the Youtube video: http://www.youtube.com/watch?v=5el1A5B-h3Q