A new constant of motion in coherent modes

The time change rate of $\varepsilon$ and $P_{\varphi}$ is given respectively by Eqs. (35) and (52), i.e.,

$$\dot{\varepsilon} = - \left( \frac{Z e}{c} \frac{\partial \ensure... ...}}} + \mu \frac{\partial B}{\partial t} + Z e \frac{\partial \phi}{\partial t},$$ (54)

$$\dot{P}_{\varphi} = \left[ \frac{Z e}{c} \left( \frac{\partial A_... ...ac{\partial B}{\partial \varphi} - Z e \frac{\partial \phi}{\partial \varphi} .$$ (55)

From Eqs. (54) and (55), we know that the energy $\varepsilon$ is conserved for motion in time independent field while $P_{\varphi}$ is conserved for motion in toroidal symmetrical field. For the motion in a toroidal symmetrical equilibrium field superposed by a coherent perturbation $h_1 (R, Z) e^{i (- n \varphi - \omega t)}$ with $n \neq 0, \omega \neq 0$, neither of $\varepsilon$ and $P_{\varphi}$ is conserved. In this case we can construct a new conservative quantity by combining $\varepsilon$ and $P_{\varphi}$. Define

$$\varepsilon' \equiv \varepsilon + \frac{\omega}{n} P_{\varphi},$$ (56)

then it is easy to verify that $d \varepsilon' / d t = 0$ when including only the contribution of the perturbation up to the order $O (h_1 / h_0)$.