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

Abstract

The underdamped harmonic oscillator is analyzed through the complex mapping $ζ= e^{-iφ}we^{-w}$ with $w = βt + iΩt$, which transforms the dynamics into a logarithmic spiral. Within this framework, the displacement extrema correspond to crossings of the imaginary axis by $ζ(t)$, yielding the explicit times $t_n = (θ- φ- π/2 + nπ)/Ω$, where $θ= \arctan(Ω/β)$. The Lambert $W$ function provides closed-form solutions $t = -β^{-1}W_k(-βA/ω_0)$ for the times at which the spiral radius attains a given threshold $A$, covering both the rising and decaying branches. The quality factor $Q = ω_0/(2β) = \tfrac{1}{2}\secθ$ is directly encoded in the ray angle $θ$ of the $(u,v)$-plane. Key geometric invariants are derived: the winding number $N_\varepsilon \approx (Q/π)\ln(2Q/\varepsilon)$ for large $Q$, the enclosed area $A = ω_0^2Ω/(8β^3) \approx Q^3$ in the lightly damped limit, and the energy decay $E(t) = E_0 e^{-ω_0 t/Q}$. 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.