= Solution
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 <Proton>[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.
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