Conservative forcing in averaged oscillator equations ($R_T=0,{\theta }_T=-⟨f(R\cos \varphi )\cos \varphi ⟩/R$)

For a weak force depending only on position, the amplitude average is a periodic total derivative and vanishes. The leading harmonic amplitude is constant, while the phase may drift. The method of multiple scales agrees with the exact conservation of energy for bounded orbits in the effective potential. A cubic position force on the right side of the oscillator equation lowers the frequency at first order when its coefficient is positive.