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The transverse transport of a charged-particle beam relative to a reference trajectory. Linear focusing gives betatron oscillations, described by Hill's equation and Courant-Snyder parameters; the beam envelope depends on its geometric emittance.

Small transverse oscillation about an accelerator's reference orbit. In a linear periodic lattice the displacement obeys Hill's equation $z^{\prime \prime }+k(s)z=0$.

Use transverse coordinates $(x,x^{\prime })$ and a signed focusing strength $K$ in Hill's equation, $x^{\prime \prime }+Kx=0$. A drift of length $\ell$ has beam transfer matrixFor a uniform focusing quadrupole magnet with $K={\kappa }^2>0$, and a defocusing one with $K=-{\kappa }^2$,The focusing plane has oscillatory trajectories, while the defocusing plane has hyperbolic growth and decay. A magnetic quadrupole focuses one transverse plane and defocuses the other; reversing its gradient interchanges their roles.

If $|K|{\ell }^2\ll 1$ and the element's internal drift can be neglected or assigned separately, its kick may be concentrated at one position. With $1/f=|K|\ell$,Both kicks preserve position and change the slope linearly with displacement. All these ideal uncoupled maps have determinant one.