Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S19 / 1 / c

For a real uncoupled, lossless transverse map, $\det M=1$. Bounded nondegenerate oscillatory motion requires complex-conjugate eigenvalues $e^{\pm i\mu }$, equivalentlyIts Courant-Snyder parameters giveThe positive quadratic form ${\gamma }_T{x}^2+2{\alpha }_{T}x{x}^{\prime }+{\beta }_T{x}^{\prime 2}$ is invariant, so repeating the cell cannot produce secular growth. At trace $\pm 2$ a nontrivial Jordan block instead produces linear growth; only the exceptional maps $M=\pm I$ remain bounded at those exact boundaries. The criterion must hold in both transverse planes. For the symmetric FODO cell, both traces are $2-d^2/f^2$, giving

For ultrarelativistic particles with fixed magnet settings, $A(\delta )=A_0(1-\delta )$ and $B(\delta )=B_0(1-\delta )$. Hence the half-trace isOff-energy motion is stable when $|C(\delta )|<1$, separately in each transverse plane. Where $\sin {\mu }_0\ne 0$, its first-order phase shift isFor equal opposite strengths,Thus positive momentum error reduces the focusing and phase advance; negative error moves toward the overfocused unstable boundary. The unexpanded symmetric stability condition is $|f_0|(1+\delta )>d/2$, with $1+\delta >0$, within the ultrarelativistic energy scaling. Its natural chromaticity per cell is $d(\mu /2\pi )/d\delta =-\tan ({\mu }_0/2)/\pi$. Away from the ultrarelativistic limit replace the displayed first-order energy error by $\delta /{\beta }_0^2$. Linearizing at a stability boundary is singular and needs separate treatment.