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arxiv: 2509.12389 · v3 · submitted 2025-09-15 · ⚛️ physics.class-ph · physics.flu-dyn

Closing a catenary loop: the lariat chain, the string shooter, and the heavy elastica

A. R. Dehadrai , J. A. Hanna This is my paper

Pith reviewed 2026-05-18 16:26 UTC · model grok-4.3

classification ⚛️ physics.class-ph physics.flu-dyn
keywords string shootercatenary loopheavy elasticadrag forcesbifurcationsgravityclosed equilibria
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0 comments X

The pith

Closed loops of axially moving strings under gravity and drag can be formed by continuing catenary solutions past vertical points.

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

The paper reviews and extends work on the equilibria of a string shooter, a type of catenary made from a steady axially moving inextensible string subject to gravity and drag. It focuses on the challenge of closing the loop by passing the string through vertical orientations, a difficulty that alters as drag increases and causes bifurcations in the system. Solutions are constructed using existing analytical results for the basic case and generated numerically when bending stiffness is included to model a heavy elastica. The authors also address global balances of linear, angular, and pseudo-momentum for these configurations. This connects string dynamics in motion to related problems like chain fountains and lariats.

Core claim

The authors show that closed loop shapes for the string shooter are achievable by implementing analytical catenary solutions and numerically extending them with bending stiffness, overcoming the inherent difficulty in continuing the shape through vertical orientations which changes with drag strength and bifurcations.

What carries the argument

The catenary configuration of a steady axially moving, inertial, inextensible, perfectly flexible string under gravity and drag, extended to closed loops and with added bending stiffness.

If this is right

  • Analytical results can be used to construct basic closed loop solutions.
  • Numerical methods generate additional solutions when bending stiffness is added.
  • The difficulty in closing the loop changes nature as the system bifurcates with increasing drag.
  • Global balances of linear, angular, and pseudo-momentum hold for the closed system.

Where Pith is reading between the lines

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

  • If these solutions are stable, they could explain observed behaviors in physical string shooting experiments.
  • Adding bending stiffness suggests that real materials with some rigidity can still form such loops.
  • Similar techniques might apply to modeling the chain fountain in closed configurations.

Load-bearing premise

The string remains steady, axially moving, inertial, inextensible, and perfectly flexible even when the configuration passes through vertical orientations and drag increases, allowing continuation of the catenary solution to close the loop.

What would settle it

Observing whether a physical string shooter maintains a closed loop when the moving string passes through a vertical segment under varying drag forces.

Figures

Figures reproduced from arXiv: 2509.12389 by A. R. Dehadrai, J. A. Hanna.

Figure 1
Figure 1. Figure 1: A string shooter in operation, and its inventor [5]. The string moves axially clockwise with speed [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Catenary loops with low, moderate, and high drag ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Catenary loops with moderate drag D = 1.2958 and launch angles θ(0) = − π 3 , 0, π 4 , π 3 . (a) Shapes. (b) Arc length s, (c) shifted tension σ − v 2 , and (d) curvature dsθ, as functions of angle θ. Curves in (d) intersect on the axis at θ = − π 2 . 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Stiff catenary loops with bending stiffness [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Stiff catenary loops with moderate drag D = 1.2958, launch angle θ(0) = π 4 , and different different return angles or stiffnesses (E = 0.01, θ(1) = −1.308π), (E = 0.001, θ(1) = −1.308π), (E = 0.01, θ(1) = −π), alongside a non-stiff (E = 0) catenary with the same drag and launch angle, and the same return angle (θ(1) = −1.308π) as two of the stiff catenaries. (a) Shapes. (b) Arc length s, (c) shifted tensi… view at source ↗
read the original abstract

We review, critique, and extend results related to the problem of closed loop shape equilibria of a string shooter, a type of catenary consisting of steady, axially moving configurations of an inertial, inextensible, perfectly flexible string in the presence of gravity and drag forces. We relate to similar problems, including the lariat (no gravity), chain fountain (not closed), and heavy \emph{elastica} (bending stiffness). We focus on the difficulty inherent to continuing a catenary through a vertical orientation, necessary to close a loop, which difficulty changes in nature as the system undergoes bifurcations with increasing drag. We construct solutions by implementing available analytical results, and numerically generate additional solutions with added bending stiffness. We briefly discuss global balances of linear, angular, and pseudo-momentum for this system.

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 / 2 minor

Summary. The manuscript reviews, critiques, and extends results on closed-loop shape equilibria of a string shooter: steady, axially moving, inertial, inextensible, perfectly flexible string configurations under gravity and drag. It connects this to the lariat (no gravity), chain fountain (not closed), and heavy elastica (with bending stiffness). The central focus is the difficulty of continuing a catenary solution through vertical orientations to close the loop and how this changes with drag-induced bifurcations. Solutions are constructed by implementing available analytical results and by numerically generating additional solutions with added bending stiffness; global balances of linear, angular, and pseudo-momentum are briefly discussed.

Significance. If the constructions hold, the paper offers a useful synthesis of prior analytical catenary results with new numerical heavy-elastica cases that address the vertical-orientation singularity and drag bifurcations. Credit is due for the explicit implementation of existing analytical solutions and the generation of additional numerical families with small bending stiffness, which directly tackles a known continuation obstacle. The discussion of global momentum balances, even if not used as a closure, adds context for the modeling assumptions of steady axial motion, inextensibility, and near-perfect flexibility.

major comments (1)
  1. [Numerical generation of solutions section] Numerical generation of solutions section: the claim that additional solutions are generated with added bending stiffness to close the loop lacks any description of the discretization, boundary conditions at the vertical passage, or convergence tests as the stiffness parameter approaches zero; without these the accuracy of the reported closed-loop shapes cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract states that prior results are 'critiqued' but the main text does not isolate specific critiques from extensions or implementations of the analytical catenary formulas.
  2. [Global balances paragraph] Global balances paragraph: the discussion of linear, angular, and pseudo-momentum balances would be clearer if accompanied by the explicit integral statements or the manner in which they are verified for the constructed solutions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive comments on our manuscript. We appreciate the positive assessment of the synthesis of analytical and numerical results. Below we address the major comment.

read point-by-point responses

  1. Referee: [Numerical generation of solutions section] Numerical generation of solutions section: the claim that additional solutions are generated with added bending stiffness to close the loop lacks any description of the discretization, boundary conditions at the vertical passage, or convergence tests as the stiffness parameter approaches zero; without these the accuracy of the reported closed-loop shapes cannot be assessed.

    Authors: We acknowledge that the current manuscript provides limited details on the numerical implementation in the section discussing solutions with added bending stiffness. To address this, in the revised version we will add a more thorough description of the numerical method. Specifically, we will detail the discretization scheme (e.g., the type of finite element or collocation method used for the heavy elastica equations), the boundary conditions applied to handle the passage through vertical orientations (leveraging the bending stiffness to avoid singularities), and include convergence studies showing the limit as the stiffness parameter tends to zero. This will allow better assessment of the reported closed-loop shapes and their relation to the analytical catenary solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript constructs solutions by implementing existing analytical catenary results from the literature and numerically generates additional heavy-elastica cases with small bending stiffness. No load-bearing step reduces a claimed prediction or first-principles result to a parameter fitted from the same data or to a self-citation whose content is itself unverified. Global momentum balances are discussed separately and are not invoked as an unverified closure. The derivation therefore remains self-contained against external benchmarks and prior independent results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on standard assumptions about string properties and steady motion stated in the abstract; no new entities are introduced.

axioms (2)
  • domain assumption The string is inertial, inextensible, and perfectly flexible.
    Explicitly stated in the abstract as the basis for the catenary model under gravity and drag.
  • domain assumption Configurations are steady and axially moving.
    Required for the equilibrium shape analysis and loop closure.

pith-pipeline@v0.9.0 · 5677 in / 1186 out tokens · 43400 ms · 2026-05-18T16:26:45.746241+00:00 · methodology

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Reference graph

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