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arxiv: 2606.00870 · v1 · pith:2JRMO3K7new · submitted 2026-05-30 · 🧮 math.DS · math-ph· math.MP· nlin.SI

From bungee to C¹ and C⁰ Hamiltonian systems and their integrability and nonintegrability

Vladimir Dragovi\'c , Borislav Gaji\'c , Bo\v{z}idar Jovanovi\'c This is my paper

Pith reviewed 2026-06-28 17:51 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MPnlin.SI
keywords Hamiltonian systemsintegrabilityLiouville-Arnold theoremC0 potentialsC1 potentialsbungee jumpingnonintegrabilitydynamical systems
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The pith

Natural Hamiltonian systems with C^0 or C^1 potentials on a hypersurface can satisfy the Liouville-Arnold theorem in prototype cases.

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

The paper introduces notions of integrability and non-integrability for natural Hamiltonian systems whose potentials are only continuous or once differentiable on a hypersurface but smooth elsewhere. Using models of bungee jumping as motivation, it derives and analyzes such systems. It provides examples where these systems are integrable according to the Liouville-Arnold theorem despite the reduced smoothness. This extends the classical theory to Hamiltonians with limited regularity across certain surfaces.

Core claim

The authors define integrability for C^0 and C^1 Hamiltonian systems and show that prototype examples exist where the Liouville-Arnold theorem applies, meaning the systems possess a sufficient number of independent integrals in involution leading to quasi-periodic motion on invariant tori, even when the potential has reduced smoothness on a hypersurface.

What carries the argument

The newly introduced notions of integrability and non-integrability for C^0 and C^1 potentials, which extend the classical definitions to allow the Liouville-Arnold theorem to hold for these systems.

If this is right

  • Integrable C^0 and C^1 systems possess a complete set of integrals that commute and are independent, leading to integrable dynamics.
  • Bungee jumping models serve as concrete physical examples of such systems.
  • Non-integrability can be established for some C^0 or C^1 systems using the new definitions.
  • The smoothness is only required away from the hypersurface, allowing for a broader class of potentials.

Where Pith is reading between the lines

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

  • This framework could be applied to other physical systems with sharp transitions or impacts.
  • Further study might reveal how the behavior changes exactly at the hypersurface in terms of conserved quantities.
  • Connections to hybrid dynamical systems or switched systems may be possible.
  • Testing the definitions on more complex potentials could validate their consistency.

Load-bearing premise

The new definitions of integrability and non-integrability for C0 and C1 Hamiltonians are consistent and accurately reflect the intended dynamical properties across the hypersurface.

What would settle it

A specific C0 or C1 Hamiltonian system that meets all the new integrability criteria but fails to exhibit the expected quasi-periodic motion on tori or lacks the required number of integrals.

Figures

Figures reproduced from arXiv: 2606.00870 by Borislav Gaji\'c, Bo\v{z}idar Jovanovi\'c, Vladimir Dragovi\'c.

Figure 1
Figure 1. Figure 1: The return map Θ+ : (x0, p0) 7−→ (x1, p1) 7−→ (x2, p2). Remark 2.1. At M± h , we have that S = x1p2−x2p1 = cos(β)ℓ|p| = cos(β)ℓ √ 2h, where β is the angle between the outgoing for M+ h and ingoing for M− h velocity p and the circle ∂D with respect to the standard orientation dx1 ∧ dx2. The sets M± h are foliated on the circles δ + h,s = M+ h ∩ {S = s}, δ− h,s = M− h ∩ {S = s} and the dynamics is simply the… view at source ↗
Figure 2
Figure 2. Figure 2: The effective potential and the phase of the reduced one￾dimensional system. is a C 1–diffeomophism between (T ∗R+∖{(r ∗ s , 0)}){r, pr} and S 1×R+{ψ (mod 2π), I}. The Hamiltonian dynamics in the new variables (ψ, I) is C∞-smooth: ˙I = 0, ψ˙ |I=I(h) = Ω(I)|I=I(h) = ∂Hs ∂I |I=I(h) = 1 ∂I/∂Hs|Hs=h = 2π Th,s , where Th,s is the period of a trajectory that belongs to the cycle γh. Here we use the fact that the… view at source ↗
Figure 3
Figure 3. Figure 3: The cylinder {H1 = h, S = s} and the torus {H2 = h, S = s} in R 4 (left) and the regular invariant torus T 2 h,s (right). The intersection are the cycles δ + h,s and δ − h,s. The intersection of the cylinder and the torus contains two circles δ + h,s and δ − h,s, which correspond to the level sets {S = s} of the return maps on M+ h and M− h (see Remark 2.1). The circles δ + h,s and δ − h,s are Poincar´e’s … view at source ↗
Figure 4
Figure 4. Figure 4: The trajectory of the C 0 -model are on the left and of the C 1 -model on the right of bungee jumping at the Verzasca Dam that starts at the origin, with the initial velocity p10 = 1.43m/s, p20 = 0. Inside the disk D, the trajectories in both cases are arcs of parabolas, represented in blue. Outside the disk, the trajectories are arcs of ellipses on the left, represented in red, while on the right are the … view at source ↗
Figure 5
Figure 5. Figure 5: On the left: a trajectory of C 0 -approximate system with a positive energy; on the right: three parabolas with nearby initial conditions staring at x0 and having the same positive energy, one of which is tangent to the boundary circle. However, the mappings Φ and Ψ are not continuous. Indeed, let P(x0, p0) = n x(t) = x0 + tp0 − 1 2 gt2 i |t ∈ R o be the parabola or a vertical ray, with the initial conditi… view at source ↗
Figure 6
Figure 6. Figure 6: An ellipse E(x0, p0) that belongs to D1, which is tan￾gent to the boundary circle ∂D (left). The corresponding solution (x(t), p(t)) given by (3.7) intersects ∆ in the points (x0, p0) ∈ ∆+, (x1, p1) ∈ ∆0, and (x2, p2) ∈ ∆−. All ellipses E(x0, p0), (x0, p0) ∈ ∆+, that are within D2, intersect ∂D in exactly two points (right) [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The trajectories of the return map Θh + in the coordinates x1 (abscissa) and S (ordinate, see Remark 2.1 to recall the definition of S) for ℓ = 1, g = 7, ρ = 5, H = h = −5.5 after 1000 iterations. we present the trajectories of the system, in the case g = 7, ρ = 5, ℓ = 1, for several initial conditions with the fixed value of the Hamiltonian H = h = −5.5 satisfying the condition (3.16) (see [PITH_FULL_IMA… view at source ↗
Figure 8
Figure 8. Figure 8: Five trajectories of the system with the C 0 -Hamiltonian (1.7), that correspond to trajectories of the return map Θ+ given in [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The effective potential for the C 0 -approximative system, the cases with |s| < √ρℓ2 , |s| = √ρℓ2 , and |s| > √ρℓ2 , respectively. (2.8) for S = s ̸= 0 and the C 0 -effective potential (4.3), and ∆φh,0 = 0 for S = 0). However, now ∆φh,s will be given in the algebraic form as well. We use the complex notation, where S(x, p) = x1p2 − x2p1 = i 2 (xp¯ − px¯) and J(x, p) = ⟨x, p⟩ = 1 2 (xp¯ + px¯). As in the C … view at source ↗
Figure 10
Figure 10. Figure 10: The invariant cylinder of the free motion (up), the res￾onant torus for the Hook potential (down, left), and C 0 -invariant torus T 2 h,s with C 0 -dynamics (down, right). The return maps Θ± are rotations of the invariant circles δ ± h,s. At the level of C 0 -gluing of two super-integrable systems, we have a similar situation as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
read the original abstract

We consider natural Hamiltonian systems with potentials that are $C^0$ or $C^1$ on a hypersurface and $C^{\infty}$-smooth in the complement and introduce and study corresponding notions of their integrabilty and non-integrability. As a motivating example, we derive and analyze models of bungee jumping. We provide prototype examples of the Liuoville-Arnol'd theorem for $C^0$ and $C^1$ Hamiltonians.

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

2 major / 1 minor

Summary. The manuscript introduces notions of integrability and non-integrability for natural Hamiltonian systems whose potentials are merely C^0 or C^1 across a hypersurface (and C^∞ elsewhere). Motivated by bungee-jumping models, it claims to supply prototype examples realizing the Liouville-Arnold theorem in this low-regularity setting.

Significance. A rigorous extension of Liouville-Arnold to C^0/C^1 Hamiltonians would be of interest for modeling mechanical systems with interfaces. The provision of explicit prototype examples, if they survive the regularity drop, would constitute a concrete contribution; however, the low-regularity setting introduces fundamental questions about flow uniqueness that must be resolved for the claim to hold.

major comments (2)
  1. [Abstract] Abstract and opening paragraph: the prototype examples of Liouville-Arnold for C^1 Hamiltonians presuppose that the common level sets of the integrals are invariant tori with well-defined dynamics. Because a C^1 potential yields only a C^0 Hamiltonian vector field, uniqueness of solutions across the hypersurface is not automatic (Peano existence holds but Lipschitz fails in general); the manuscript must either prove uniqueness on the interface or supply a weaker notion of integrability that does not rely on unique orbits.
  2. [Definitions section] Definitions of integrability/non-integrability (presumably §2 or §3): these notions must be shown to be mathematically consistent with the flow on the full phase space; if the vector field is merely continuous, the statement that level sets are foliated by tori becomes ill-posed without additional arguments establishing that trajectories cannot branch at the hypersurface.
minor comments (1)
  1. [Abstract] Abstract: 'Liuoville-Arnol'd' is misspelled (should be Liouville-Arnold).

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed report. We address the major comments regarding uniqueness of solutions and consistency of definitions for low-regularity Hamiltonian systems below. We believe these points can be resolved with clarifications and additions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraph: the prototype examples of Liouville-Arnold for C^1 Hamiltonians presuppose that the common level sets of the integrals are invariant tori with well-defined dynamics. Because a C^1 potential yields only a C^0 Hamiltonian vector field, uniqueness of solutions across the hypersurface is not automatic (Peano existence holds but Lipschitz fails in general); the manuscript must either prove uniqueness on the interface or supply a weaker notion of integrability that does not rely on unique orbits.

    Authors: We agree that uniqueness of solutions must be addressed explicitly for the C^1 case. In the manuscript, the bungee-jumping models are constructed such that the vector field is continuous and the energy conservation allows unique continuation across the interface by solving the ODE in each smooth region and matching at the hypersurface. However, to make this rigorous, we will add a dedicated lemma proving local uniqueness of solutions for these natural Hamiltonians with C^1 potentials. This will be included in a revised version of the paper. revision: yes

  2. Referee: [Definitions section] Definitions of integrability/non-integrability (presumably §2 or §3): these notions must be shown to be mathematically consistent with the flow on the full phase space; if the vector field is merely continuous, the statement that level sets are foliated by tori becomes ill-posed without additional arguments establishing that trajectories cannot branch at the hypersurface.

    Authors: The definitions in §2 are formulated in terms of the existence of sufficiently many independent integrals in involution whose common level sets are compact and connected, and we verify in the prototype examples that the flow is uniquely determined by the Hamiltonian equations in the smooth regions and extended continuously. To address the potential branching issue, we will include an argument showing that for natural Hamiltonians, the trajectories are uniquely determined by the position and velocity matching at the interface due to the form of the equations of motion. This ensures the level sets are indeed invariant under the flow. revision: yes

Circularity Check

0 steps flagged

No circularity: new integrability notions for C0/C1 Hamiltonians introduced independently

full rationale

The paper introduces fresh definitions of integrability and non-integrability for C0 and C1 Hamiltonians on a hypersurface (with smooth complement) and constructs prototype examples from bungee-jumping models. No load-bearing step reduces by definition, by fitting, or by self-citation chain to its own inputs; the Liouville-Arnold prototypes are built directly from the newly stated notions rather than presupposing them. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

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