text: The magnetic moment of a gyrating particle is
  which respects special relativity.  is the relativistic Lorentz factor,  is the rest mass,  is the velocity perpendicular to the magnetic field, and  is the magnitude of the magnetic field. 
   is a constant of the motion to all orders in an expansion in , where  is the rate of any changes experienced by the particle, e.g., due to collisions or due to temporal or spatial variations in the magnetic field. Consequently, the magnetic moment remains nearly constant even for changes at rates approaching the gyrofrequency. When  is constant, the perpendicular particle energy is proportional to , so the particles can be heated by increasing , but this is a "one-shot" deal because the field cannot be increased indefinitely. It finds applications in magnetic mirrors and magnetic bottles.
  There are some important situations in which the magnetic moment is not invariant:
  ; Magnetic pumping: If the collision frequency is larger than the pump frequency, μ is no longer conserved. In particular, collisions allow net heating by transferring some of the perpendicular energy to parallel energy.
  ; Cyclotron heating: If B is oscillated at the cyclotron frequency, the condition for adiabatic invariance is violated, and heating is possible. In particular, the induced electric field rotates in phase with some of the particles and continuously accelerates them.
  ; Magnetic cusps: The magnetic field at the center of a cusp vanishes, so the cyclotron frequency is automatically smaller than the rate of any changes. Thus the magnetic moment is not conserved, and particles are scattered relatively easily into the loss cone.