= Energy balance in a uniformly rotating frame
{title2=$\dot E=(\partial_t\phi)_{r'}$}
For uniform <angular velocity> $\omega$, potential $\phi$, and additional <forces> doing no work in the rotating frame,
$$
E=\frac m2|v'|^2+\phi-\frac m2|\omega\times r|^2,
\qquad \dot E=(\partial_t\phi)_{r'}.
$$
The <Coriolis force> does no work; the <centrifugal potential> accounts for the remaining apparent <force>. This energy is conserved precisely when the potential has no explicit time dependence in the rotating coordinates. An inertial time-independent <conservative force> need not meet that condition unless its potential is also invariant under the frame's rotation. The balance prevents applying a rotating-frame conservation law to a nonaxisymmetric stationary inertial potential.
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