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arxiv: 2605.31456 · v1 · pith:XYBSELRPnew · submitted 2026-05-29 · 🧮 math-ph · math.MP

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

Arpan Sharma , Bhargava Jogi , Ken Roberts , Muralikrishna Molli , S.R. Valluri This is my paper

Pith reviewed 2026-06-28 19:55 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords damped harmonic oscillatorLambert W functionlogarithmic spiralquality factorcomplex mappingwinding numberenergy decay
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The pith

The Lambert W function supplies closed-form times for the underdamped oscillator to reach any amplitude threshold A via a logarithmic spiral mapping.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps the underdamped harmonic oscillator to a logarithmic spiral in the complex plane using ζ = e^{-iφ} w e^{-w} with w = β t + i Ω t. This yields explicit times for displacement extrema at imaginary-axis crossings and, via the Lambert W function, closed-form solutions for the times when the spiral radius equals any chosen threshold A. The quality factor appears directly as Q = (1/2) sec θ where θ is the spiral ray angle, and the same geometry produces expressions for winding number, enclosed area, and energy decay. A reader cares because the approach replaces numerical root-finding or approximation with exact special-function expressions and a visual picture of the decay process.

Core claim

The underdamped harmonic oscillator is transformed into a logarithmic spiral by the complex mapping ζ = e^{-iφ} w e^{-w} with w = β t + i Ω t. Displacement extrema occur when ζ(t) crosses the imaginary axis, at the explicit times t_n = (θ - φ - π/2 + nπ)/Ω with θ = arctan(Ω/β). The Lambert W function then gives the times at which the spiral radius reaches any threshold A as t = -β^{-1} W_k(-β A / ω_0) for appropriate branches k. The quality factor is encoded as Q = (1/2) sec θ, the winding number is N_ε ≈ (Q/π) ln(2Q/ε) for large Q, the enclosed area is A = ω_0² Ω / (8 β³) ≈ Q³ in the light-damping limit, and energy decays as E(t) = E_0 e^{-ω_0 t / Q}. Three experimental methods for extracti

What carries the argument

The complex mapping ζ = e^{-iφ} w e^{-w} that converts the damped-oscillator trajectory into the geometry of a logarithmic spiral whose radius and crossings correspond to physical amplitude and extrema.

If this is right

  • Times of all amplitude extrema and any chosen threshold crossings are given by closed-form expressions rather than numerical root finding.
  • Quality factor can be read directly from the measured ray angle θ or obtained by counting spiral turns, with the latter stable even when successive peaks differ by tiny amounts.
  • Winding number scales as (Q/π) ln(2Q/ε) and enclosed area scales as Q³ in the lightly damped regime.
  • Energy decays exponentially at the exact rate ω_0 / Q without additional approximation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same spiral construction could be applied to other linear second-order systems whose solutions involve decaying exponentials times oscillations.
  • The Lambert-W expression for threshold times could be used inside event-detection routines in simulation codes to avoid interpolation errors.
  • Plotting experimental trajectories in the mapped (u, v) plane might reveal damping anomalies that are hidden in ordinary time-series plots.

Load-bearing premise

The damped-oscillator solution must be exactly underdamped so that the complex mapping produces a true logarithmic spiral whose radius is proportional to the physical displacement amplitude.

What would settle it

Integrate the differential equation numerically for chosen β, ω_0 and A, record the times at which |x(t)| equals A, and verify whether those times agree with the values computed from t = -β^{-1} W_k(-β A / ω_0) to within integration tolerance.

Figures

Figures reproduced from arXiv: 2605.31456 by Arpan Sharma, Bhargava Jogi, Ken Roberts, Muralikrishna Molli, S.R. Valluri.

Figure 1
Figure 1. Figure 1: Spring–mass–damper system. Applying Newton’s second law yields the second-order homogeneous linear ODE: M d 2y dt2 + C dy dt + Ky = 0. (2) Assuming a solution of the form e mt, the characteristic equation is Mm2 + Cm + K = 0. (3) with roots m1,2 = − C 2M ± 1 2M p C2 − 4MK. (4) 2.2. Damping Regimes 2.2.1. Overdamping (C 2 − 4KM > 0) The solution can be written as y(t) = e −( C 2M )t [PITH_FULL_IMAGE:figure… view at source ↗
Figure 2
Figure 2. Figure 2: The black curves show the locus satisfying (21). For [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometric interpretation of Q in the uv-plane (u = βt, v = Ωt). Each ray corresponds to a fixed Q, with slope tan θ = Ω/β and Q = 1 2 sec θ. The horizontal ray (θ = 0◦ ) is the critically damped limit Q = 1/2; as θ → 90◦ , Q → ∞. Compare the ray angles here with the spiral figures: a larger θ produces more spiral turns. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Spiral ζ(t) = e −iφwe−w for four underdamped parameter sets. Colour encodes time (colour bar at right); the red dashed line marks Re(ζ) = 0 and its crossings give displacement extrema. (a) β = 0.57, Ω = 2.17, φ = −37.2 ◦ : Q ≈ 2.1; spiral completes ∼2–3 turns with multiple imaginary-axis crossings visible. (b) β = 0.57, Ω = 2.17, φ = 78.5 ◦ : same Q ≈ 2.1 as (a); changing φ rotates the spiral without alter… view at source ↗
Figure 5
Figure 5. Figure 5: A damped oscillator with time stamp from 0 to 10 secs . The green dot indicates the start of the motion of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spiral ζ(t) = e −iφwe−w for the critically damped (Q = 0.5), and overdamped (Q = 0.3) harmonic oscillator (φ = π/4). Colour encodes time. The red dashed line marks Re(ζ) = 0. In the critically and over-damped cases the spiral collapses to a straight line, confirming aperiodic motion. 6.2. Total Mechanical Energy E = 1 2My˙ 2 + 1 2Ky2 . Substituting (67)–(68): E(t) = 1 2MD2 e −2βth β 2 c 2 θ + 2βΩcθsθ + Ω2 … view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper analyzes the underdamped harmonic oscillator by mapping its solution to a logarithmic spiral in the complex plane via the transformation ζ = e^{-iφ} w e^{-w} with w = βt + iΩt. It derives explicit times t_n for displacement extrema as imaginary-axis crossings, obtains closed-form times t = -β^{-1} W_k(-β A / ω_0) via the Lambert W function for the spiral radius reaching threshold A (covering rising and decaying branches), encodes the quality factor as Q = (1/2) sec θ where θ is the ray angle, derives invariants including winding number N_ε ≈ (Q/π) ln(2Q/ε), enclosed area A ≈ Q^3, and energy decay E(t) = E_0 e^{-ω_0 t / Q}, and compares three experimental methods for Q (logarithmic decrement, ray-angle, turn-counting), claiming robustness of the latter for high-Q systems.

Significance. If the claimed correspondences hold, the geometric framework unifies the damped oscillator with complex analysis and the Lambert W function, yielding parameter-free geometric invariants and a potentially robust high-Q measurement technique via spiral turn counting. The explicit Lambert W expressions and Q-angle relation would constitute a novel interpretive tool with possible pedagogical and experimental value.

major comments (1)
  1. [Abstract and mapping definition] Abstract and mapping definition (ζ = e^{-iφ} w e^{-w}, w = βt + iΩt): the central claim that spiral-radius properties correspond directly to physical amplitude thresholds (enabling Lambert W solutions for times when radius attains threshold A) is undermined because |ζ(t)| = |w| e^{-Re(w)} = ω_0 t e^{-β t}, whereas the physical envelope of x(t) is strictly proportional to e^{-β t}. Solving |ζ(t)| = A therefore addresses t e^{-β t} = const (Lambert W with rising/decaying branches from the t prefactor), which has no counterpart in physical amplitude thresholds (which reduce to a simple logarithm without Lambert W). This severs the asserted direct correspondence between radius crossings/properties and physical extrema or thresholds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful analysis and for identifying the mismatch in the claimed correspondence. We address the major comment below and agree that revisions are required.

read point-by-point responses

  1. Referee: [Abstract and mapping definition] Abstract and mapping definition (ζ = e^{-iφ} w e^{-w}, w = βt + iΩt): the central claim that spiral-radius properties correspond directly to physical amplitude thresholds (enabling Lambert W solutions for times when radius attains threshold A) is undermined because |ζ(t)| = |w| e^{-Re(w)} = ω_0 t e^{-β t}, whereas the physical envelope of x(t) is strictly proportional to e^{-β t}. Solving |ζ(t)| = A therefore addresses t e^{-β t} = const (Lambert W with rising/decaying branches from the t prefactor), which has no counterpart in physical amplitude thresholds (which reduce to a simple logarithm without Lambert W). This severs the asserted direct correspondence between radius crossings/properties and physical extrema or thresholds.

    Authors: We agree with the referee's calculation and observation. Direct verification confirms |ζ(t)| = ω₀ t e^{-β t}, introducing an extraneous linear factor of t relative to the physical envelope e^{-β t}. Consequently, the Lambert W expressions solve for radius thresholds on |ζ(t)| and do not furnish closed-form times for physical amplitude thresholds (which are indeed simple logarithms). The imaginary-axis crossings for extrema remain valid as stated, but the linkage of radius properties to physical amplitude thresholds is not supported. We will revise the abstract, the Lambert W section, and all related claims to remove or correct the asserted direct correspondence, explicitly distinguishing the geometric radius from the physical envelope. This constitutes a major revision. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations follow directly from the introduced mapping and standard identities

full rationale

The paper introduces the mapping ζ = e^{-iφ} w e^{-w} with w = βt + iΩt as a framework that transforms the oscillator into a logarithmic spiral, then derives t_n from imaginary-axis crossings and the Lambert W expression from the radius |ζ| = ω0 t e^{-β t} reaching threshold A. These steps are explicit consequences of the chosen mapping and the definitions θ = arctan(Ω/β), Q = ω0/(2β). The identity Q = (1/2) sec θ follows at once from cos θ = β/ω0. No parameters are fitted to data and then relabeled as predictions, no self-citations are invoked as load-bearing uniqueness theorems, and no result is shown to equal its own input by construction. The energy decay, winding number, and area expressions are likewise obtained by direct substitution of the standard underdamped solution into the new coordinates. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper draws on standard mathematical axioms and the known analytic solution of the damped harmonic oscillator without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption The underdamped solution is x(t) = A e^{-β t} sin(Ω t + φ)
    Basis for the mapping to the spiral.
  • standard math Lambert W function satisfies W(z) exp(W(z)) = z
    Used to solve for t in the radius equation.

pith-pipeline@v0.9.1-grok · 5861 in / 1376 out tokens · 35243 ms · 2026-06-28T19:55:48.609302+00:00 · methodology

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