Geometric Analysis of the Damped Harmonic Oscillator via the Lambert W Function
The underdamped harmonic oscillator is analyzed through the complex mapping $\zeta = e^{-i\varphi}we^{-w}$ 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 $t_n = (\theta - \varphi - \pi/2 + n\pi)/\Omega$, 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 $Q = \omega_0/(2\beta) = \tfrac{1}{2}\sec\theta$ is directly encoded in the ray angle $\theta$ of the $(u,v)$-plane. Key geometric invariants are derived: the winding number $N_\varepsilon \approx (Q/\pi)\ln(2Q/\varepsilon)$ 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.