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

 (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

 (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

 (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

 (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