A variational principle and the classical and quantum mechanics of the damped harmonic oscillator

A variational principle is presented, by means of which the equation of motion of the damped harmonic oscillator is found. Starting from this variational principle a systematic reformulation of the classical mechanics leads us to a Hamilton–Jacobi equation with an additional term, which is proportional to the action. The quantization of this Hamilton–Jacobi equation, following a method originally due to Schrödinger, gives the Langevin–Schrödinger equation.