2. Hill's Equations

Hill's equations; linearised second order differential equations for the transverse variables $x$ and $y$ in dipole and quadrupole fields within an accelerator are :

\begin{equation}x(s)'' + \left( k(s) + \frac{1}{\rho(s)^2} \right) x(s) ~=~ 0,\end{equation}
and
\begin{equation}y(s)'' - k(s) y ~=~ 0.\end{equation}

where $\rho = const$ and $k=const$ . Hills equations may be written compactly;

\begin{equation}x''+ K \cdot x ~=~ 0.\end{equation}

Where $K = k, K = (k + \frac{1}{\rho^2})$; if we write K for the constant and assume that it is constant over the accelerator, these equations appear similar to that of the a harmonic oscillator.