  Energy equipartition is appropriate only for nonrelativistic classical
mechanics, but has only limited relevance for a relativistic theory such as
classical electrodynamics. In this article, we discuss harmonic-oscillator
thermal equilibrium from three different perspectives. First, we contrast the
thermal equilibrium of nonrelativistic mechanical oscillators (where point
collisions are allowed and frequency is irrelevant) with the equilibrium of
relativistic radiation modes (where frequency is crucial). The Rayleigh-Jeans
law appears from applying a dipole-radiation approximation to impose the
nonrelativistic mechanical equilibrium on the radiation spectrum. In this
discussion, we note the possibility of zero-point energy for relativistic
radiation, which possibility does not arise for nonrelativistic
classical-mechanical systems. Second, we turn to a simple electromagnetic model
of a harmonic oscillator and show that the oscillator is fully in radiation
equilibrium (which involves all radiation multipoles, dipole, quadrupole, etc.)
with classical electromagnetic zero-point radiation, but is not in equilibrium
with the Rayleigh-Jeans spectrum. Finally, we discuss the contrast between the
flexibility of nonrelativistic mechanics with its arbitrary potential functions
allowing separate scalings for length, time, and energy, with the
sharply-controlled behavior of relativistic classical electrodynamics with its
single scaling connecting together the scales for length, time, and energy. It
is emphasized that within classical physics, energy-sharing, velocity-dependent
damping is associated with the low-frequency, nonrelativistic part of the
Planck thermal radiation spectrum, whereas acceleration-dependent radiation
damping is associated with the high-frequency adiabatically-invariant and
Lorentz-invariant part of the spectrum corresponding to zero-point radiation.
