Time-dependent damped harmonic oscillators

Abstract

In this work we analyze the solutions of the equations of motions of Lane-Emden oscillators, which are associated with a mass m(t) = m0tα , α > 0. These systems are damped harmonic oscillators with a time-dependent damping coefficient, γ(t) = <img src="/img/revistas/rbef/v35n4/a11img01.jpg" width="17" height="15" align="absmiddle" /> We obtain analytical expression for x(t), <img src="/img/revistas/rbef/v35n4/x_ponto.jpg" width="12" height="12" align="baseline" />(t) = v(t), e p(t) = m(t)<img src="/img/revistas/rbef/v35n4/x_ponto.jpg" width="12" height="12" align="baseline" /> for α = 2 and α = 4. We discuss the differences between the expressions for the Hamiltonian and the mechanical energy for time-dependent systems. We also compared our findings with the results for the well-known Caldirola-Kanai oscillators.

damped oscillators; Frobenius' method; equation of motion