Robinsonʹs radiation damping sum rule: Reaffirmation and extension

Robinsonʹs radiation damping sum rule is one of the classic theorems of accelerator physics. Recently Orlov has claimed to find serious flaws in Robinsonʹs proof of his sum rule. In view of the importance of the subject, I have independently examined the derivation of the Robinson radiation damping sum rule. Orlovʹs criticisms are without merit: I work through Robinsonʹs derivation and demonstrate that Orlovʹs criticisms violate well-established mathematical theorems and are hence not valid. I also show that Robinsonʹs derivation, and his damping sum rule, is valid in a larger domain than that treated by Robinson himself: Robinson derived his sum rule under the approximation of a small damping rate, but I show that Robinsonʹs sum rule applies to arbitrary damping rates. I also display more concise derivations of the sum rule using matrix differential equations. I also show that Robinsonʹs sum rule is valid in the vicinity of a parametric resonance.