Courant-Snyder parameters
= Courant-Snyder parameters
{c}
{title2=$\alpha_T=-\beta'/2,\quad\gamma_T=(1+\alpha_T^2)/\beta$}
The positive function $\beta(s)$ and the displayed associated parameters describe a stable linear beam ellipse. A trajectory with invariant $\epsilon$ obeys $\epsilon=\gamma_Tz^2+2\alpha_Tzz'+\beta z'^2$ and has phase advance $\phi'=1/\beta$. Writing $\omega=\sqrt\beta$ gives $\omega''+k(s)\omega=1/\omega^3$; the coefficient of $k$ is $\omega$, not one.