-\omega {\mathbf{a}}^{\prime} \sin(\tau +{\mathbf{\varphi}})  - \omega {\mathbf{a}} {\mathbf{\varphi}}^{\prime} \cos(\tau +{\mathbf{\varphi}})  - \omega^2 {\mathbf{a}} \cos(\tau +{\mathbf{\varphi}})$$ and substituting $\ddot{{\mathbf{z}}}$ into (\[eq:mvt\]): $$\label{eq:Slow2}
    -\omega {\mathbf{a}}^{\prime} \sin(\tau +{\mathbf{\varphi}})  - \omega {\mathbf{a}} {\mathbf{\varphi}}^{\prime} \cos(\tau +{\mathbf{\varphi}})  - \omega^2 {\mathbf{a}} \cos(\tau +{\mathbf{\varphi}}) + g({\mathbf{a}} \cos (\tau +{\mathbf{\varphi}}),-\omega {\mathbf{a}} \sin(\tau +{\mathbf{\varphi}}))={\mathbf{0}}$$

Finally, solving equations (\[eq:Slow1\]) and (\[eq:Slow2\]) for slow flow variables variations ${\mathbf{a}}^{\prime}$ and ${\mathbf{\varphi}}^{\prime}$ we have the following system:

\[eq:SlowStand\] $${\mathbf{a}}^{\prime}=\dfrac{\sin(\tau +{\mathbf{\varphi}})}{\omega} \left(G({\mathbf{a}},{\mathbf{\varphi}},\tau)-\omega^2 {\mathbf{a}}\cos(\tau +{\mathbf{\varphi}})\right)$$ $${\mathbf{a}}{\mathbf{\varphi}}^{\prime}=\dfrac{\cos(\tau +{\mathbf{\varphi}})}{\omega} \left(G({\mathbf{a}},{\mathbf{\varphi}},\tau)-\omega^2 {\mathbf{a}}\cos(\tau +{\mathbf{\varphi}})\right)$$

where $G({\mathbf{a}},{\mathbf{\varphi}})$ is obtained by substituting ${\mathbf{z}}$ and $\dot{{\mathbf{z}}}$ in $g$.

The system (\[eq:SlowStand\]) is now in standard form and can be averaged over the fast time scales $\tau$ with the slow flow variables ${\mathbf{a}}$ and ${\mathbf{\varphi}}$ being taken as constants.

\[eq:SlowMoy\] $${\mathbf{a}}^{\prime} = \dfrac{1}{\omega} \dfrac{1}{2\pi}\displaystyle \int_0^{2\pi}G({\mathbf{a}},{\mathbf{\varphi}},\psi)\sin \psi d\psi$$ $${\mathbf{a}}{\mathbf{\varphi}}^{\prime} = \dfrac{1}{\omega} \left[ \dfrac{1}{2\pi}\displaystyle \int_0^{2\pi} G({\mathbf{a}},{\mathbf{\varphi}},\psi) \cos\psi d\psi-\dfrac{\omega^2}{2} {\mathbf{a}}\right]$$

The averaged problem defined by the differential system (\[eq:SlowMoy\]) provides approximate solutions with the particular form defined by (\[eq:solapprox\]). The approximation is made on the form of the nonlinear terms in $g$ which are assumed (due to averaging) to be proportional to the harmonic functions $\cos \psi$ and $\sin \psi$.

This formalism will be used in the following section to derive particular solutions to different problems, including 