Motion in time of a swinging Atwood machine

Giorgio Carizi

REVIEW 3 major objections 4 minor 6 references

Motion in time of a swinging Atwood machine

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims an exact closed-form time law for the teardrop-heart orbits of the swinging Atwood machine, giving period $T=4\sqrt{2E}$ that is independent of launch angle.

desk verdict A real extension of Tufillaro's teardrop-heart result, but the exactness claims need to be relabeled as asymptotic (r0→0) before the paper is ready for publication. read the letter →

arxiv 2507.00058 v1 pith:YOU626NI submitted 2025-06-12 physics.class-ph nlin.SI

Giorgio Carizi This is my paper · ORCID

classification physics.class-phnlin.SI

keywords swingingAtwoodmachineteardrop-heartorbitsexactsolutionfictitioustimeextendedHamiltonianisochronousmotionclosed-formlawregularization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims to supply the missing time dependence for the teardrop-heart orbits of the swinging Atwood machine, a two-mass pulley system in which one mass swings in a vertical plane. Using a fictitious-time transformation, the author separates the motion into two independent quartic oscillators and integrates each one in elementary closed form. The resulting expressions give the radius and angle as functions of real time, and the period is derived as $T=4\sqrt{2E}$. Because for these orbits the energy $E=2\dot r_0^2$, the period depends only on the radial launch speed and not on the launch angle, so the motion is isochronous in the launch angle.

What carries the argument

The central mechanism is the fictitious-time regularization with an extended Hamiltonian: real time $\varpi$ is traded for a fictitious time $\tau$ via $d\varpi=(\xi^2+\eta^2)d\tau$, and the new Hamiltonian $\mathcal{H}=(\xi^2+\eta^2)(H-E)$ is formed. This converts the fixed-energy motion into zero-pseudo-energy motion of two independent oscillators, each in one of the variables $\xi,\eta$ with quartic potential. The load-bearing quantity is the separation constant between the two oscillators: once it is set to zero, each oscillator equation is integrable in terms of exponentials and logarithms, yielding the closed forms for $\xi^2(\tau)$, $\eta^2(\tau)$, and $\varpi(\tau)$.

What would settle it

Compute that separation constant for a finite starting radius, say $r_0=0.1$, from the paper's formulas: if it is nonzero, the exact solution does not hold. Alternatively, numerically integrate the original equations of motion and compare the measured period and trajectory with the paper's closed forms; any difference beyond numerical error at the order $r_0^2$ would confirm the approximation.

Watch

Extended reading notes

Core claim

The central claim is that the complete motion in real time for the teardrop-heart orbits can be written in closed form. Starting from the orbit variables $(\xi,\eta)$ in which the Hamiltonian becomes two decoupled quartic oscillators, the paper introduces the fictitious time $\tau$ by $d\varpi=(\xi^2+\eta^2)d\tau$ and the extended Hamiltonian $\mathcal{H}=(\xi^2+\eta^2)(H-E)$. The separation of $\mathcal{H}$ yields a constant of motion; setting that constant to zero from the initial conditions allows each oscillator to be integrated explicitly as $\xi^2(\tau)=2AE\,e^{-\sqrt{2E}\tau}/\bigl(1+Ae^{-\sqrt{2E}\tau}\bigr)^2$ and similarly for $\eta^2(\tau)$. Substituting into the definitions of $r$ and $\theta$, and integrating $d\varpi/d\tau$, gives the time-parametrized trajectory and the relation $\varpi(\tau)=\sqrt{2E}\bigl[f(A)-g(A,\tau)+f(B)-g(B,\tau)\bigr]$. In the limit $\tau\to\infty$ the real time accumulates to $2\sqrt{2E}$, interpreted as half the period, and since $E=2\dot r_0^2$ for these orbits, the period is independent of the initial launch angle.

Load-bearing premise

The solution assumes that a certain separation constant of the motion is exactly zero once the launch conditions are plugged in; that is true only in the limit of zero starting radius, so for a real finite starting radius the closed-form expressions are approximations rather than exact results.

Editorial extensions

If this is right

The period of teardrop-heart orbits is $T=4\sqrt{2E}=8|\dot r_0|$, so all such orbits launched with the same radial speed return in the same time regardless of launch angle.
The solution provides complete real-time parametrizations $r(\varpi)$ and $\theta(\varpi)$, filling the gap left by the orbit equation alone.
The explicit map $\varpi(\tau)$ assigns a real time to every point on the orbit, so the full motion can be animated rather than just traced.
As $\tau$ goes to infinity the motion returns to the origin in the finite real time $2\sqrt{2E}$, giving a well-defined semi-period.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The exactness claim is fragile: expanding the initial conditions at finite $r_0$ gives a nonzero separation constant of order $r_0^2$, so the closed forms are exact only in the limit $r_0\to0$ and approximate for the finite starting radius used in the numerical example.
The same fictitious-time separation could be tried on other special orbits of the $\mu=3$ swinging Atwood machine, since the decoupling that makes the integration work depends on the quartic structure of the transformed Hamiltonian.
The launch-angle isochrony suggests a hidden symmetry of the teardrop family; deriving the period formula from a conservation law would be a natural testable extension.
A direct numerical integration of the original equations for finite $r_0$ would quantify the error in the closed-form solution and identify the regime where the approximation is good.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper gives the explicit τ-domain solution and real-time quadrature for Tufillaro's teardrop-heart orbits, which is a genuine extension. But the 'exact' label is doing too much work: the solution is exact only in the r0→0 limit, and Eq. (16) is a limit statement, not a finite-radius identity. After a revision that says this plainly, it deserves publication in a specialized venue.

What's new: Tufillaro solved the orbit equation for μ=3 SAM but left the trajectory (motion in time) unresolved. This paper supplies ξ²(τ), η²(τ), and ϖ(τ) via the Poincaré/Levi-Civita extended Hamiltonian and quadrature of dϖ=(ξ²+η²)dτ. The algebraic form looks right in the intended limit; the decoupled quartic oscillators integrate to the rational-exponential forms (13)–(14), and Eq. (15) correctly accumulates real time. No parameter fitting, no circular reasoning—just a standard regularization applied to a known integrable family.

The soft spots are real. Plugging (6) into (9) gives I2=O(r0²), not identically zero, because ξ̇0 has a 1/√r0 piece that cancels the −Eξ² term only at leading order. So for any finite r0, the exact solution is elliptic, not (13)–(14). The paper starts with 'infinitesimal' ε, so the limit is arguably the intended regime, but the abstract and Eq. (16) drop the qualifier. Similarly, f(α)=α/(1+α)<1, so lim ϖ = √(2E)[f(A)+f(B)] equals 2√(2E) only as A,B→∞, i.e., again as r0→0. These are not trivial details; they change whether the period formula is an equality or an asymptote. The text also says A and B are 'very well approximate' by 2E/ξ0², which is itself a finite-radius approximation.

There are also presentation issues: the integration steps from (13') to (13) and the derivation of (15) are sketched, and the text has enough OCR artifacts to make the formulas harder to check than they need to be. But the core derivation is reproducible.

Bottom line: the paper is a solid, if narrow, contribution to a classical mechanics niche. It should go to peer review, with a request to restate the exactness claims as asymptotic and give the limit conditions explicitly. Anyone working on SAM or Levi-Civita regularizations will want to cite the corrected version.

Referee Report

3 major / 4 minor

Summary. The paper proposes a closed-form solution for the time evolution of the swinging Atwood machine (SAM) with mass ratio μ=3, for initial conditions near the origin (the so-called teardrop-heart orbits). Using a Poincaré-type time transformation and separating the regularized Hamiltonian into two decoupled oscillators, the author derives expressions for ξ^2(τ), η^2(τ), and the real time ϖ(τ) in Eqs. (13)-(15), and from them concludes that the semi-period is T/2 = 2√(2E), independent of the launch angle (Eq. (16)). The abstract claims that this is an exact, isochronous solution for the motion in time.

Significance. If the exactness claim were correct, the paper would provide a valuable closed-form time parametrization for a family of orbits of a mechanical system with a non-trivial regularized Hamiltonian, complementing Tufillaro's orbit equation. The derivation is elegant, and the algebra connecting the I2=0 system to the closed forms is sound; in particular, Eqs. (13)-(14) do satisfy the first-order equations derived from the separated Hamiltonian with I2=0, and Eq. (15) integrates Eq. (7) correctly. However, the exactness claim is not supported: the separation constant I2 vanishes only in the limit r0→0, and the semi-period formula (16) is a limit rather than an exact equality for finite r0. With appropriate qualifications as an asymptotic solution valid for infinitesimal starting radius, the result could be a useful complement to existing work; as stated, the central claim needs to be revised.

major comments (3)

[Sec. 2, Eqs. (9)-(10) and the sentence 'Plugging the initial conditions (6) into Eqs. (9) and (10) gives Ι2 = 0'] A direct expansion of Eq. (9) using the initial data (6) gives I2 = O(r0^2) with a coefficient that is generically non-zero; for the parameters of Fig. 2, for instance, I2 is of order 10^-17 rather than zero. Hence the closed forms (13)-(14) are exact solutions of the I2=0 subsystem, not of the original SAM with finite r0; they become exact only in the limit r0→0. This contradicts the abstract's claim of an exact solution for teardrop-heart orbits, and the manuscript should explicitly state the asymptotic nature of the result.
[Sec. 3, Eq. (16)] Taking the τ→∞ limit in Eq. (15) gives lim ϖ = √(2E)[f(A)+f(B)], not 2√(2E) unless A,B→∞. Since f(α)=α/(1+α)<1 for finite α, the equality in Eq. (16) holds only in the limit r0→0 (where A≃2E/ξ0^2 and B≃2E/η0^2 both diverge). For the finite r0=10^-8 used in Fig. 2, the semi-period differs from 2√(2E) by a correction of order r0/E, and the exact finite-r0 period should be written as √(2E)[f(A)+f(B)]. Consequently, the claimed isochrony with respect to launch angle is established only asymptotically, not as an exact property.
[Sec. 3, text after Eq. (13)] The statement that A 'is very well approximate by the quantity 2E/ξ0^2' is another indication that the solution is not exact as written: A is in fact fixed by the initial condition ξ0^2 = 2AE/(1+A)^2, which has an exact solution, and the approximation 2E/ξ0^2 is valid only when A is large. The paper should either solve for A and B exactly or present the solution explicitly as an asymptotic one in the limit r0→0; the current phrasing leaves the exactness claim ambiguous.

minor comments (4)

[Eq. (13')] The logarithmic expression is typeset in a way that makes the intended parentheses unclear; please rewrite it so that the argument is unambiguous.
[Reference [4]] The author name is listed as 'C.L. C.L. Siegel'; the duplication should be removed.
[Throughout] The constant of motion is written variously as 'Ι2' (Greek iota) and 'I2'; please use a single notation consistently.
[Abstract and Introduction] The abstract and introduction should explicitly state that the solution is obtained in the infinitesimal-radius limit ε→0, rather than presenting it as an exact solution for a finite initial radius; this would prevent a misleading reading of the result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the time solution is integrated from the Hamiltonian with boundary-fitted constants; the finite-radius approximation is a correctness issue, not a circular one.

full rationale

The derivation chain is self-contained. The paper starts from the Lagrangian, applies the regularizing transformation (7)-(8), separates the Hamilton-Jacobi constants (9)-(10), and integrates the resulting oscillator equations to obtain the closed forms (13)-(14). The constants A and B are fixed by the initial conditions (6), which is standard solution of an initial-value problem, not a fit to the predicted period or orbit. The semi-period (16) is obtained by integrating the time transformation (15), and the isochrony statement follows from the initial-condition energy relation E=2ṙ0² rather than being imposed as the conclusion. No load-bearing step relies on a self-citation: the cited Tufillaro and Poincaré results are external and are used to set up the transformation, not to justify the final prediction. The text does contain a genuine mathematical limitation: the claim that plugging (6) into (9)-(10) gives I2=0 is only exact in the limit r0→0, and the text itself says A is 'very well approximate' by 2E/ξ0² while using ε=10^-8 in Figure 2. This makes the closed-form solution and Eq. (16) asymptotic rather than exact at finite radius, but that is an approximation/correctness concern, not a circular reduction of the prediction to its inputs. No parameter is fitted to the quantity being predicted, and no conclusion is assumed to derive itself.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central solution depends on four quantities chosen or approximated from initial conditions: the integration constants A,B, the energy E, and the constant I2 set to zero. No new physical entities are introduced. The key approximation is I2=0, valid only in the r0→0 limit, which is load-bearing for the decoupled oscillators and the period formula.

free parameters (4)

A (ξ integration constant) = ≈ 2E/ξ0²
Set so that ξ²(0)=ξ0²; exact value is the large root of ξ0² = 2EA/(1+A)², approximated as 2E/ξ0² in Eq. (13).
B (η integration constant) = ≈ 2E/η0²
Set so that η²(0)=η0²; approximated as 2E/η0² in Eq. (14).
E (energy) = 2 ṙ0²
In the limit r0→0, the Hamiltonian energy becomes 2 ṙ0²; used throughout the solution. For finite r0 there are corrections of order r0 and r0².
I2 (constant of motion) = 0
Obtained by plugging initial conditions (6) into (9)-(10) in the limit r0→0; exact I2 is O(r0²) for finite r0.

assumptions (4)

domain assumption Tufillaro's transformation (2)-(3) and the Hamiltonian (4) correctly describe the μ=3 SAM.
The paper takes these from Ref. [2] without proof; the entire solution relies on them.
standard math The Poincaré extended Hamiltonian formalism: dynamics in real time ϖ with H=E are equivalent to dynamics in fictitious time τ with ℋ=0 via dϖ=(ξ²+η²)dτ.
Standard regularization theory (Refs. [3-6]); used to convert the singular system into two quartic oscillators.
ad hoc to paper For teardrop-heart orbits the initial radius r0 is infinitesimal and the constant I2 vanishes in the limit r0→0.
The paper states I2=0 after plugging (6) into (9)-(10), but this is only exact in the limit; finite r0 gives I2=O(r0²).
ad hoc to paper The constants A and B are very well approximated by 2E/ξ0² and 2E/η0².
Used to write the closed forms (13)-(14) and the period (16); the exact A,B solving the initial conditions are not given.

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Cite this review

Pith. "Pith review of Motion in time of a swinging Atwood machine." pith.science (2026). https://pith.science/paper/YOU626NI

@misc{pith2026250700058,
  author       = {Pith},
  title        = {Pith review of: Motion in time of a swinging Atwood machine},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOU626NI}},
  note         = {Machine review of arXiv:2507.00058}
}

read the original abstract

An exact solution for the motion in time of the swinging Atwood machine is presented. The trajectories of the so called teardrop-heart orbits are obtained considering an extended Hamiltonian and a time transformation first introduced by Poincare. The motion is isochronous with respect to the launch angle.

Figures

Figures reproduced from arXiv: 2507.00058 by the authors.

**Figure 1.** Swinging Atwood machine In the (𝜉, 𝜂) variables the Hamiltonian of the system takes the form 𝐻 = 1 (𝜉 2+𝜂 2) [ 1 8 (𝑝𝜉 2 + 𝑝𝜂 2 ) +2(𝜉 4 + 𝜂 4 )] (4) where 𝑝𝜉 = 4𝜉̇(𝜉 2 + 𝜂 2 ) 𝑝𝜂 = 4𝜂̇(𝜉 2 + 𝜂 2 ) (5) In the new variables the initial conditions read 𝜉0 = √𝑟0√1 + 𝑠𝑖𝑛(𝜃0⁄2) (𝑝𝜉) 0 = 4𝜉̇ 0 (𝜉0 2 +𝜂0 2 ) (6) 𝜂0 = ±√𝑟0√1 − 𝑠𝑖𝑛(𝜃0⁄2) (𝑝𝜂) 0 = ±4𝜂̇0 (𝜉0 2 + 𝜂0 2 ) where 𝜉̇ 0 = 𝜉0𝑟0̇ 2𝑟0 + 𝑟0𝜃̇ 0 4𝜉0 𝑐𝑜𝑠 ( 𝜃0 2 ) and 𝜂̇0 =… view at source ↗

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Smiles and Teardrops

N.B. Tufillaro, "Smiles and Teardrops", B.A. thesis, Reed College, Portland, OR, (1982)

work page 1982
[2]

Teardrop and heart orbits of a swinging Atwood's machine

N. B. Tufillaro, " Teardrop and heart orbits of a swinging Atwood's machine", Am. J. Phys. 62 (3), 231-233 (1994)

work page 1994
[3]

Sur la résolution qualitative du problème restreint des trois corps

T. Levi -Civita, "Sur la résolution qualitative du problème restreint des trois corps", Acta Mathematica, 30, 305-327 (1906)

work page 1906
[4]

Lectures on Celestial Mechanics

C.L. C.L. Siegel and J.K. Moser, "Lectures on Celestial Mechanics", Springer-Verlag, Berlin Heidelberg New York, 1971

work page 1971
[5]

Dynamics and integrability of the swinging Atwood machine generalisations

W. Szuminski and A.J. Maciejewski, "Dynamics and integrability of the swinging Atwood machine generalisations", Nonlinear Dyn. 110, 2101 –2128 (2022)

work page 2022
[6]

Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems

J. Struckmeier, "Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems", arXiv:2304.09633v1. Teardrop-heart orbit (heart) 𝜛 = 𝜛(𝜏) Eq. (15) 𝑟 = 𝑟(𝜛) Eq. (2) 𝜃 = 𝜃(𝜛) Eq. (3) Figure 2. Motion in time of the swinging mass for initial conditions (𝑟 = 10−8, 𝑟̇0 = 0.3, 𝜃0 = 3𝜋 4⁄ , 𝜃0̇ = 0.1). 𝑇 2 = 1.2 0.0 0.2 0...

work page Pith review arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Smiles and Teardrops

N.B. Tufillaro, "Smiles and Teardrops", B.A. thesis, Reed College, Portland, OR, (1982)

work page 1982

[2] [2]

Teardrop and heart orbits of a swinging Atwood's machine

N. B. Tufillaro, " Teardrop and heart orbits of a swinging Atwood's machine", Am. J. Phys. 62 (3), 231-233 (1994)

work page 1994

[3] [3]

Sur la résolution qualitative du problème restreint des trois corps

T. Levi -Civita, "Sur la résolution qualitative du problème restreint des trois corps", Acta Mathematica, 30, 305-327 (1906)

work page 1906

[4] [4]

Lectures on Celestial Mechanics

C.L. C.L. Siegel and J.K. Moser, "Lectures on Celestial Mechanics", Springer-Verlag, Berlin Heidelberg New York, 1971

work page 1971

[5] [5]

Dynamics and integrability of the swinging Atwood machine generalisations

W. Szuminski and A.J. Maciejewski, "Dynamics and integrability of the swinging Atwood machine generalisations", Nonlinear Dyn. 110, 2101 –2128 (2022)

work page 2022

[6] [6]

Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems

J. Struckmeier, "Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems", arXiv:2304.09633v1. Teardrop-heart orbit (heart) 𝜛 = 𝜛(𝜏) Eq. (15) 𝑟 = 𝑟(𝜛) Eq. (2) 𝜃 = 𝜃(𝜛) Eq. (3) Figure 2. Motion in time of the swinging mass for initial conditions (𝑟 = 10−8, 𝑟̇0 = 0.3, 𝜃0 = 3𝜋 4⁄ , 𝜃0̇ = 0.1). 𝑇 2 = 1.2 0.0 0.2 0...

work page Pith review arXiv