Double Pendulum as a Topological Bridge

In the paper's words

The double pendulum serves as the paradigmatic model for the transition from integrability to chaos, yet its potential as a rigorous bridge between classical nonlinear dynamics and quantum open systems remains underutilized. By treating the system’s phase space as a topologically tiled manifold under 2π periodicity, we establish a structural isomorphism between classical Lyapunov instability and quantum decoherence rates. Leveraging recent advances in open quantum system dynamics (Hernández et al., 2025) and topological critical phases (Yu et al., 2026), we demonstrate that classical “flip” events correspond to topological winding number transitions that drive quantum state delocalization.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. A pivotal hypothesis in quantum foundations connects this process directly to classical chaos: the rate of entropy production (decoherence rate) $\Gamma$ in a chaotic quantum system is determined by the sum of the positive Lyapunov exponents $\lambda$ of its classical limit, such that $\Gamma \propto \sum \lambda_i^+$ (Zurek & Paz, 1994).

   from 2.4 Decoherence and the

2. Due to the computational prohibitive cost of diagonalizing the full double pendulum Hamiltonian at high quantum numbers ($N \gt 10,000$), we employed a Random Matrix Theory (RMT) proxy validated against established spectral properties of the system (Perotti, 2004).

   from 3.4 Quantization Protocol

3. The sensitivity analysis indicates that near the threshold of instability, infinitesimal perturbations ($\delta \sim 10^{-5}$) can switch a trajectory from librational to rotational.

   from 4.4 Fractal Dimension

4. Specifically, the chaotic states (where $\lambda \approx 1.06$) decohere rapidly, with the entropy production rate matching the Kolmogorov-Sinai entropy of the classical attractor (Zurek & Paz, 1994).

   from 5.3 Decoherence Rates

How it is done

1. Research Objectives Despite the rich literature on both the double pendulum and quantum chaos, significant gaps prevent a complete synthesis of these fields.

   from 1.4 Gap Analysis and

2. To rigorously test this thesis, we employ a hybrid methodology combining high-precision symplectic integration of classical trajectories with exact diagonalization of the quantized Hamiltonian.

   from 1.6 Methodological Approach

3. This symplectic structure ensures the conservation of phase space volume (Liouville’s theorem), creating an incompressible flow that prevents trajectories from overlapping.

   from 2.1 Classical Hamiltonian

4. Mapping Dictionary To formalize the structural isomorphism between these domains, we construct a “Rosetta Stone” dictionary.

   from 2.5 The Rosetta Stone:

What would falsify it

1. Numerical simulations reveal that the fractal geometry of stability islands quantitatively predicts the resilience of quantum states to environmental coupling, suggesting a novel “decoherence-free” encoding scheme based on Kolmogorov-Arnold-Moser (KAM) tori.

   from Abstract

2. In the double pendulum, this predicts that “flip” regions—where $\lambda$ is maximal—will undergo the fastest decoherence, effectively “collapsing” into classical particle-like behavior, while the librational islands remain coherent.

   from 2.4 Decoherence and the

3. Topological Transitions The “flip” of the outer pendulum—defined as a full $2\pi$ rotation ($\theta_2$ crossing $\pm \pi$)—emerges as the discrete observable quantifying topological transitions.

   from 4.2 Flip Dynamics and

Where the claim stops

1. Deep in the quantum regime (small quantum numbers), the concept of a “trajectory” and “local winding number” becomes ill-defined due to the uncertainty principle.

   from 6.4 Limitations of the

2. However, these classical descriptions often overlook the system’s profound capacity to model phase transitions that transcend classical mechanics.

   from 1.1 The Double

3. However, the precise mechanism by which this transition occurs remains a subject of intense theoretical debate.

   from 1.2 The Core

Where to read

1. Isomorphisms26 words · 1 min
2. Abstract152 words · 1 min
3. Keywords13 words · 1 min
4. 1.0 Introduction1,753 words · 8 min
5. 2.0 Theoretical1,410 words · 6 min
6. 3.0 Methodology:1,117 words · 5 min
7. 4.0 Results I: The1,139 words · 5 min
8. 5.0 Results II: The Quantum1,176 words · 5 min
9. 6.0 Discussion: The Universal1,044 words · 5 min
10. 7.0 Conclusion692 words · 3 min
11. Appendices787 words · 3 min