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It is pointed out that an electric charge oscillating in a one-dimensionalpurely-harmonic potential is in detailed balance at its harmonics with aradiation bath whose energy $U_{rad}$ per normal mode is linear in frequency$\omega$, $U_{rad}=const\times\omega,$ and hence is Lorentz invariant, as seemsappropriate for relativistic electromagnetism. The oscillating charge is NOT inequilibrium with the Rayleigh-Jeans spectrum which arises from energy-sharingequipartition ideas which are valid only in nonrelativistic mechanics. Here weexplore the contrasting behavior of harmonic oscillators connected to baths inclassical mechanics and electromagnetism. It is emphasized that modern physicstext are in error in suggesting that the Rayleigh-Jeans spectrum corresponds tothe equilibrium spectrum of random classical radiation, and in ignoringLorentz-invariant classical zero-point radiation which is indeed a classicalequilibrium spectrum.

Conflict Between Classical Mechanics and Electromagnetism: The Harmonic Oscillator in Equilibrium with a Bath