Noting:

\[\tau = t-t_s \qquad \phi = \varphi - \varphi_s \qquad \delta = \frac{p-p_s}{p_s} \qquad w = W-W_s\]

From panofsky equation, we have the discrete energy gain per turn defined by the gap voltage and RF phase:

\[\Delta W = qV_g\cos\varphi\]

In relative coordinates we accordingly have:

\[\Delta w = \Delta W - \Delta W_s = qV_g(\sin\varphi-\sin\varphi_s)=qV_g g(\phi)\]

From $\Delta w/\Delta t \approx \dot{w}$ and letting $g(\phi) = \sin\varphi-\sin\varphi_s$:

\[\dot{w} = \frac{qV_g}{T_s}g(\phi)\]

Changing variables from $\phi = h\omega_s\tau$, we can write:

\[\boxed{\dot{\tau} = \kappa w \qquad \dot{w} = \frac{qV(\tau)}{T_s}}\]

Where:

\[\kappa = \frac{\eta}{\beta_s^2E_s} \qquad V(\tau) = V_g g(h\omega_s \tau)\]

And in discretized form we have:

\[\boxed{\Delta w = qV(\tau) \qquad \Delta\tau = \kappa T_s w}\]

We can derive our EOM from the following hamiltonion:

\[\boxed{H = \frac{1}{2}\kappa w^2-\frac{q}{T_s}\int V(\tau)d\tau}\]

For an RF voltage source(s) of harmonic $h$ and gap voltage $V_h$, we have:

\[V(\tau) = \sum V_h g_h(\tau)\]

Where:

\[\begin{aligned} g(\phi) &= \sin\varphi-\sin\varphi_s\cr &=\sin(\phi+\varphi_s)-\sin\varphi_s\cr &= \sin\phi\sin\varphi_s+\cos\phi\cos\varphi_s-\sin\varphi_s\cr &= \sin\varphi_s(\sin\phi-1)+\cos\varphi_s\cos\phi \end{aligned}\]

Therefore:

\[G(\phi) = \int g(\phi) d\phi = -\sin\varphi_s(\cos\phi+\phi)+\cos\varphi_s\sin\phi\]

So our hamiltonion simplifies to:

\[H = \frac{1}{2}\kappa w^2 - \frac{q}{T_s}G(\tau)\]

For small $\phi$ and $\phi = h\omega_s\tau$:

\[g(\phi) \approx \cos\varphi_s\phi\]

Therefore:

\[\ddot{\tau} - \frac{\eta}{\beta_s^2E_s}\frac{qV_g}{T_s}\cos\varphi_sh\omega_s \tau = 0\]

Or:

\[\ddot{\tau}+\Omega_s^2\tau=0 \qquad \ddot{w}+\Omega_s^2w=0\]

where:

\[\boxed{\Omega_s^2 = -\frac{h\omega_s^2\eta}{\beta_s^2E_s} \frac{qV_h}{2\pi}\cos\varphi_s}\]