Generalized Lagrangian and conservation law for the damped harmonic oscillator

An explicit construction of the time-dependent Lagrangian for a damped harmonic oscillator is given. The connection between this Lagrangian with the time-independent one developed by Morse and Feshbach is established. Various methods of deriving a first integral for the equation of motion are given. It is shown that the results developed in this paper by direct means are in agreement with those obtained through the use of group-theoretic methods, such as invariance principles and Noether's theorem, which are mathematically more advanced approaches