  Kohn's theorem places strong constraints on the cyclotron response of Fermi
liquids. Recent observations of a doping dependence in the cyclotron mass of
La$_{2-x}$Sr$_x$CuO$_4$ (Legros et al., Phys. Rev. B 106, 195110 (2022)) are
therefore surprising because the cyclotron mass can only be renormalized by
large momentum umklapp interactions which are not expected to vary
significantly with doping. We show that a version of Kohn's theorem continues
to apply to disorder-free non-Fermi liquids with a critical boson near zero
momentum. However, marginal Fermi liquids arising from a spatially random
Yukawa coupling between the electrons and bosons do give rise to significant
corrections to the cyclotron mass which we compute. This is the same theory
which yields linear-in-temperature resistivity and other properties of strange
metals at zero fields (Patel et al., Science 381, 790 (2023)).
