Editing Adiabatic invariant (section)

== Plasma physics ==

In [[plasma physics]] there are three adiabatic invariants of charged-particle motion.

=== The first adiabatic invariant, μ ===
The '''magnetic moment''' of a gyrating particle is
<math display="block">
\mu = \frac{\gamma m_0 v_\perp^2}{2B},
</math>
which respects special relativity.<ref>{{cite book |last1=Longair |first1=Malcolm S. |title=High Energy Astrophysics |date=2011 |publisher=Cambridge University Press |location=Cambridge |isbn=978-0-521-75618-1 |page=182 |edition=3rd}}</ref> <math>\gamma</math> is the relativistic [[Lorentz factor]], <math>m_0</math> is the rest mass, <math>v_\perp</math> is the velocity perpendicular to the magnetic field, and <math>B</math> is the magnitude of the magnetic field. 

<math>\mu</math> is a constant of the motion to all orders in an expansion in <math>\omega/\omega_c</math>, where <math>\omega</math> 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 <math>\mu</math> is constant, the perpendicular particle energy is proportional to <math>B</math>, so the particles can be heated by increasing <math>B</math>, but this is a "one-shot" deal because the field cannot be increased indefinitely. It finds applications in [[magnetic mirror]]s and [[magnetic bottle]]s.

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 [[Magnetic mirror|loss cone]].

=== The second adiabatic invariant, ''J'' ===
The '''longitudinal invariant''' of a particle trapped in a [[magnetic mirror]],
<math display="block">
J = \int_a^b p_\parallel \,ds,
</math>
where the integral is between the two turning points, is also an adiabatic invariant. This guarantees, for example, that a particle in the [[magnetosphere]] moving around the Earth always returns to the same line of force. The adiabatic condition is violated in '''transit-time magnetic pumping''', where the length of a magnetic mirror is oscillated at the bounce frequency, resulting in net heating.

=== The third adiabatic invariant, Φ ===
The total magnetic flux <math>\Phi</math> enclosed by a drift surface is the third adiabatic invariant, associated with the periodic motion of mirror-trapped particles drifting around the axis of the system. Because this drift motion is relatively slow, <math>\Phi</math> is often not conserved in practical applications.