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For slowly accelerated paraxial motion, $z^{\prime }\simeq p_z/p_s$ decreases as longitudinal momentum grows while transverse canonical phase-space area is conserved. The normalized emittance stays constant in the ideal limit, so the geometric emittance scales inversely with longitudinal momentum. This is kinematic damping rather than dissipation of canonical area.

The area occupied by a beam in transverse position-momentum or position-angle phase space, with a stated area or rms convention. Geometric emittance uses trajectory angle, while normalized emittance removes the ideal kinematic damping due to acceleration.

For a paraxial orbit $z^{\prime }\simeq p_z/p_s$, 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 area becomes a smaller area in the coordinates $(z,z^{\prime })$. This is adiabatic damping of geometric emittance, not loss of canonical phase-space volume.

The invariant normalized emittance isHere ${\beta }_{\mathrm{rel}}$ 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.

Lossless Hamiltonian focusing can reduce a beam's transverse position width at the expense of momentum spread, but cannot reduce its occupied canonical phase volume. Coupling to the longitudinal direction can redistribute projected emittance without reducing total emittance. True cooling requires exporting entropy to additional degrees of freedom or discarding particles: stochastic feedback, electron cooling or radiative damping coupled to an external environment provide possible mechanisms, with very different efficiency for high-energy protons. Collimation shrinks the accepted distribution at the cost of intensity. Acceleration also reduces angular divergence at fixed normalized emittance, an apparent geometrical cooling rather than a violation of Liouville's theorem. The apparatus and radiation cannot be omitted from the accounting when the bunch is no longer an isolated Hamiltonian subsystem.