Next: Adiabatic Invariants
Up: Charged Particle Motion
Previous: Invariance of Magnetic Moment
An adiabatic invariant is an approximation to a more fundamental type of
invariant known as a Poincaré invariant (Hazeltine and Waelbroeck 2004). A Poincaré invariant
takes the form
![$\displaystyle {\cal I} = \oint_{C(t)} {\bf p}\cdot d{\bf q},$](img365.png) |
(2.71) |
where all points on the closed curve
in phase-space move
according to the equations of motion.
In order to demonstrate that
is a constant of the motion, we
introduce a periodic variable
parameterizing the points on the curve
.
The coordinates of a general point on
are thus written
and
. The rate of change of
is then
![$\displaystyle \frac{d{\cal I}}{dt} =\oint\left(p_i\,\frac{\partial^2 q_i}{\part...
...l s} +\frac{\partial p_i}{\partial t} \frac{\partial q_i}{\partial s}\right)ds.$](img371.png) |
(2.72) |
Let us integrate the first term by parts, and then use Hamilton's
equations of motion to simplify the result (Goldstein, Poole, and Safko 2002). We obtain
![$\displaystyle \frac{d{\cal I}}{dt} =\oint\left( - \frac{\partial q_i}{\partial ...
...al s}+\frac{\partial H}{\partial q_i} \frac{\partial q_i}{\partial s}\right)ds,$](img372.png) |
(2.73) |
where
is the Hamiltonian for the motion.
The integrand is now seen to be the total derivative of
along
.
Because the Hamiltonian is a single-valued function, it follows that
![$\displaystyle \frac{d{\cal I}}{dt} =-\oint\frac{d H}{ds}\,ds =0.$](img375.png) |
(2.74) |
Thus,
is indeed a constant of the motion.
Next: Adiabatic Invariants
Up: Charged Particle Motion
Previous: Invariance of Magnetic Moment
Richard Fitzpatrick
2016-01-23