Constraint dependence of longitudinal and transverse radiation comparisons ($P_{\perp }/P_{\parallel }={\gamma }^{-2}$ at fixed $a$, ${\gamma }^2$ at fixed $F$)

For one particle at a fixed Lorentz factor, the Liénard radiated-power formula gives $P_{\parallel }=C_q{\gamma }^6{a}_{\parallel }^2$ and $P_{\perp }=C_q{\gamma }^4{a}_{\perp }^2$, where $C_q=q^2/(6\pi {ϵ}_0{c}^3)$. Consequently equal coordinate acceleration magnitudes make transverse motion radiate less by ${\gamma }^2$. Equal applied laboratory force magnitudes instead give $a_{\parallel }=F/(m{\gamma }^3)$ and $a_{\perp }=F/(m\gamma )$, so $P_{\perp }/P_{\parallel }={\gamma }^2$. These opposite orderings concern distinct constraints and are consistent. An energy budget without a time, length, force, or acceleration constraint cannot by itself select a unique minimum-loss accelerator design.