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Next: Adiabatic Invariants Up: Charged Particle Motion Previous: Invariance of Magnetic Moment

Poincaré Invariants

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

$\displaystyle {\cal I} = \oint_{C(t)} {\bf p}\cdot d{\bf q},$ (2.71)

where all points on the closed curve $ C(t)$ in phase-space move according to the equations of motion.

In order to demonstrate that $ {\cal I}$ is a constant of the motion, we introduce a periodic variable $ s$ parameterizing the points on the curve $ C$ . The coordinates of a general point on $ C$ are thus written $ q_i = q_i(s,t)$ and $ p_i=p_i(s,t)$ . The rate of change of $ {\cal I}$ is then

$\displaystyle \frac{d{\cal I}}{dt} =\oint\left(p_i\,\frac{\partial^2 q_i}{\part...
...l s} +\frac{\partial p_i}{\partial t} \frac{\partial q_i}{\partial s}\right)ds.$ (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

$\displaystyle \frac{d{\cal I}}{dt} =\oint\left( - \frac{\partial q_i}{\partial ...
...al s}+\frac{\partial H}{\partial q_i} \frac{\partial q_i}{\partial s}\right)ds,$ (2.73)

where $ H({\bf p}, {\bf q}, t)$ is the Hamiltonian for the motion. The integrand is now seen to be the total derivative of $ H$ along $ C$ . Because the Hamiltonian is a single-valued function, it follows that

$\displaystyle \frac{d{\cal I}}{dt} =-\oint\frac{d H}{ds}\,ds =0.$ (2.74)

Thus, $ {\cal I}$ is indeed a constant of the motion.


next up previous
Next: Adiabatic Invariants Up: Charged Particle Motion Previous: Invariance of Magnetic Moment
Richard Fitzpatrick 2016-01-23