= Solution
For a paraxial orbit $z'\simeq p_z/p_s$, where $p_z$ is transverse <momentum> and $p_s$ longitudinal <momentum>. Acceleration mainly increases $p_s$, so a given transverse kick produces a smaller angular deviation. In an ideal matched accelerator without scattering or radiation, the conserved transverse canonical <phase space>[phase-space] area becomes a smaller area in the coordinates $(z,z')$. This is <adiabatic damping of geometric emittance>, not loss of canonical <phase space>[phase-space] volume.
The invariant <normalized emittance> is
$$
\epsilon_n=\beta_{\rm rel}\gamma_{\rm rel}\epsilon
=\frac{p_s}{m}\epsilon,
\qquad
\boxed{\epsilon\propto p_s^{-1}.}
$$
Here $\beta_{\rm rel}$ is speed in units of $c$, distinct from the lattice beta function. Neglecting dilution mechanisms, each acceleration stage thus supplies a beam with smaller angular spread and <geometric emittance> for injection into the next stage.
A chain of increasingly energetic <particle accelerators> lets each ring operate over a manageable range of <magnetic rigidity> and radio-frequency settings. It also avoids injecting a large-emittance low-energy beam directly into the small-aperture high-field main ring. Higher injection <energy> reduces space-charge effects and improves capture and transmission. Emittance dilution, mismatching and collective instabilities can oppose the ideal damping; acceleration alone does not reduce <normalized emittance>.
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