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A linear second-order ordinary differential equation with periodic coefficient. For accelerator beam optics, $s$ measures distance along the reference orbit and $k(s)$ is the local transverse focusing strength.

Maps act on column vectors, so the first element appears at the right of the product. Starting immediately before the focusing kick, one full cell hasMultiplying givesIt has $\det M_x=1$, as required for this lossless linear beam optics. The other transverse plane experiences the opposite quadrupole signs, soThe two planes have different envelopes but the same trace and therefore the same stable phase-advance magnitude for this equal-strength doublet.

Substituting the electron Lorentz factor in the supplied energy-loss formula givesWithout replenishment, the electron's momentum decreases. The reference dipoles then bend it more tightly, producing a displaced orbit. In linear beam optics, $x=D_x\delta$ and $y=D_y\delta$, where $\delta =\Delta p/p\simeq -U_0/E\simeq -1.11\times 10^{-3}$ per uncompensated turn. In an ideal planar ring the horizontal dispersion is nonzero and the orbit shifts horizontally, inward where $D_x>0$. There is no corresponding vertical closed-orbit shift when $D_y=0$. Real coupling or vertical dispersion can change that idealization.

With current $I=0.500$ A, the number of circulating particles is $N=I{T}_{\mathrm{rev}}/e$, so the radiated power isThe $2\mu \mathrm{s}$ revolution time cancels when using beam current. Radio-frequency cavities restore the lost energy longitudinally, with synchronous gain $e{V}_{\mathrm{RF}}\sin {\phi }_s=U_0$ in the sine-phase convention. Their RF power system must supply at least this beam power, in addition to its own losses. Radiation damping and quantum excitation determine beam emittances; they should not be confused with the dispersive closed-orbit shift from a momentum change.