= Constraint dependence of longitudinal and transverse radiation comparisons
{title2=$P_\perp/P_\parallel=\gamma^{-2}$ at fixed $a$, $\gamma^2$ at fixed $F$}
For one particle at a fixed <Lorentz factor>, the <Liénard radiated-power formula> gives $P_\parallel=C_q\gamma^6a_\parallel^2$ and $P_\perp=C_q\gamma^4a_\perp^2$, where $C_q=q^2/(6\pi\epsilon_0c^3)$. 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.
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