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It is generally considered that systems with friction are not part of Hamiltonian dynamics, but this is not always the case. Show that a (nonrelativistic) damped harmonic oscillator can be described by a Hamiltonian (and by a Lagrangian), with the implication that Liouville’s theorem applies here. Consider motion in coordinate x of a particle of mass m with equation of motion, mẍ+ βx ̇ + kx = 0, or x ̈ + αx ̇ + ω20x = 0, (1) where α = β/m and ω20 = k/m. Comment on the root-mean square emittance of a “bunch ” of noninteracting particles each of which obeys eq. (1). Deduce two independent constants of the motion for a single particle. Hint: Consider first the case of zero spring constant k.