On the Lagrangian and Hamiltonian description of the damped linear   harmonic oscillator
  Using the modified Prelle- Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms. 
V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan