Geometric Analysis of the Damped Harmonic Oscillator via the Lambert W Function

Abstract

The underdamped harmonic oscillator is analyzed through the complex mapping $\zeta =e^{-i\phi }w{e}^{-w}$ with $w=\beta t+i\Omega t$, which transforms the dynamics into a logarithmic spiral. Within this framework, the displacement extrema correspond to crossings of the imaginary axis by $\zeta (t)$, yielding the explicit times $t_n=(\theta -\phi -\pi /2+n\pi )/\Omega$, where $\theta =\arctan (\Omega /\beta )$. The Lambert $W$ function provides closed-form solutions $t=-{\beta }^{-1}{W}_k(-\beta A/{\omega }_0)$ for the times at which the spiral radius attains a given threshold $A$, covering both the rising and decaying branches. The quality factor $Q={\omega }_0/(2\beta )=\frac{1}{2}\sec \theta$ is directly encoded in the ray angle $\theta$ of the $(u,v)$-plane. Key geometric invariants are derived: the winding number $N_{\epsilon }\approx (Q/\pi )\ln (2Q/\epsilon )$ for large $Q$, the enclosed area $A={\omega }_0^2\Omega /(8{\beta }^3)\approx Q^3$ in the lightly damped limit, and the energy decay $E(t)=E_0{e}^{-{\omega }_{0}t/Q}$. Three methods for determining $Q$ from experimental data are compared: logarithmic decrement, ray-angle measurement, and spiral turn counting. The turn-counting method proves particularly robust for high-$Q$ systems, where successive amplitude peaks differ by tiny fractions. The framework unifies classical damped oscillations with complex analysis and special functions.