Abstract: The differential equation of damped harmonic oscillator （DHO） is transformed into an equivalent self-adjoint form by using a direct method. Two Lagrangians and Hamihonians of DHO are constructed and the Noether conserved quantities for the DHO are obtained.

Key words: damped harmonic oscillator, inverse problem of Lagrange mechanics, Lagrangian function, Hamihonian function, Noether conserved quantity