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

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

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

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

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

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