Magnetic moment and other invariants of Lorentz System It's a standard fact that the quantity $\mu = v_{\perp}^{2}/B$ is approximately conserved in the system $d_{t}\mathbf{v} = \gamma (\mathbf{v}\times\mathbf{B})$. The paper Magnetic Moment to Second Order for Axisymmetric Static Field by C. Gardner claims that this is actually just the first term in the series expansion of a function ("the" magnetic moment) whose time derivative is exactly zero, but he only finds the second term (which is a huge mess.) This sort of thing fascinates me, and I have a few questions:


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*Lets call this exact magnetic moment $M$; it has the property that $d_{t}M = 0$. We know that $\mu$ is an adiabatic invariant, but $M$ is, in some sense, "more" invariant than that, so what kind of function is it?

*What guarantee is there that $M$ won't just be a function of the kinetic energy and some conserved momenta (which have time derivatives equal to zero?) In other words, how do we know that $M$ is independent of invariants we already know?

*This is an old paper. Has anyone figured out $M$ since then, or has any more progress been made?
 A: As I understand it, the whole problem can be envisioned in terms of and applied to a charged particle moved through a plasma. In this case you have both electric and magnetic fields which exert forces on the charge, with both a translational and a rotational effect, where the latter is due to the fact that a magnetic field causes the particle to engage in circular motion. See the following image (from Wikipedia) which might aid in understanding the situation:
 
As you can see, the motion of the particle, e.g. the blue trajectory, consists of both circular motion on a trajectory of a certain radius (called the "gyroradius") and linear motion of the center of rotation, the "gyrocentre" or "guiding centre". 
But how is this related to a series expansion of the magnetic moment of the particle? 
In the case that the angular frequency (gyrofrequency) is high, i.e. we have rapid circular motion, and the electric and magnetic fields are either constant in time or slowly changing, one can simplify the problem in terms of a series expansion in the gyroradius. This is due to the fact that a large gyrofrequency corresponds to a small gyroradius. If we analyze the equations of motion to lowest order, i.e. vanishing gyroradius, we end up with the known expression for the magnetic moment of a charged particle. This quantity is a constant of motion, it is time-independent. But as shown by Kruskal, this result holds to any order. 
Note that the expansion in small gyroradius is equivalent to an expansion in the ratio of mass and charge, since the two quantities are proportional to each other.
So to answer your first question: both quantities describe magnetic moment, but one is the limit in a certain approximation. Thus, there is no fundamental difference in those invariances.
As for your second question, I doubt that there is a guarantee. However, judging and extrapolating from the form of the coefficients appearing in the first and second order expressions, it does not seem as if there is anything beautiful or elegant to expect.
Regarding your third question: I have not come across anything, but I cannot say that I am familiar with the whole literature. Nevertheless, if you really want to know if there has been any progress in that direction, it might be worth skimming through the list of papers citing the work of Gardner (14) and Kruskal (230). This might take some time, though.  
