A Damped Oscillator as a Hamiltonian System

1 Problem 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, m¨x + β ˙ x + kx = 0, or¨x + α ˙ x + ω 2 0 x = 0, (1) where α = β/m and ω 2 0 = 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. When the spring constant k is zero there is no potential energy, so the spirit of Lagrange and Hamilton is to consider the kinetic energy T = m ˙ x 2 /2. The equation of motion (1) can be written in the manner of Lagrange as 1 d dt ∂T ∂ ˙ x + α ∂T ∂ ˙ x = 0. (2) This form can be written more compactly as d dt ∂T ∂ ˙ x = 0 , where T = T e αt. Hence, a Lagrangian for this case is L = T , and the canonical momentum p conjugate to coordinate x is p = ∂L ∂ ˙ x = m ˙ x e αt = p mech e αt , (4) where p mech = m ˙ x is the ordinary mechanical momentum. The Hamiltonian for this case is then H = ˙ xp − L = L = m ˙ x 2 2 e αt. 1 See sec. 2a of [1].