benchmarks/dynamical_systems/ibm_double_pendulum/description.md

IBM Double Pendulum Chaotic Dataset — Lagrangian, Hamiltonian and equations of motion

Background

A double pendulum is a two-degree-of-freedom mechanical system: a second pendulum hung from the tip of a first pendulum. Despite having four state variables (two angles and two angular velocities) and a Lagrangian that fits in one line, it is a canonical physical example of deterministic chaos: two initially near-identical launches diverge exponentially, and no closed-form solution for θ(t) exists beyond the small-angle linearisation.

Asseman, Kornuta & Ozcan (IBM Research AI, NeurIPS 2018 MDSD workshop) built a physical double pendulum, filmed it with a high-speed camera, and released 21 video runs together with tracked marker positions per frame as the Double Pendulum Chaotic Dataset. They argue that the broader spatio-temporal prediction literature has over-fit to simulated physics, and that real apparatus like this one is the right benchmark for chaotic dynamics.

This entry turns the dataset into a symbolic-regression benchmark by framing three ground-truth formulas that any textbook-trained physicist (or LLM) will recognise:

1. eom_theta1 — the Euler–Lagrange equation of motion for θ̈₁.
2. eom_theta2 — the Euler–Lagrange equation of motion for θ̈₂.
3. hamiltonian — the total mechanical energy H = T + V.

All three use only (θ₁, θ₂, ω₁, ω₂) as inputs.

Physical setup (Asseman et al. 2018, Fig. 2b and Tab. 2c)

quantity	value	source
arm 1 length l₁ (pivot → first datum)	91 mm	Fig. 2b
arm 2 length l₂ (first datum → second)	70 mm	Fig. 2b
marker diameter	19 mm each (three markers)	Fig. 2b
camera frame rate	400 Hz	Tab. 2c
frame exposure	90 µs	Tab. 2c
image resolution	480 × 480 px, 3 channels	Tab. 2c
camera focal length / distance	50 mm / 2 m	§2.2
bob masses m₁, m₂	not reported	—
pivot friction / damping model	not reported	—

The three 19 mm fiducial markers are colour-coded: red at the pivot, green at the end of the upper arm, blue at the end of the lower arm. IBM extracts marker positions by 5× upscaling the image, drawing matched templates with scikit-image, and locating them by OpenCV template cross-correlation; each recorded pixel coordinate is therefore 5× the raw pixel position, preserving sub-pixel precision.

Each run lasts roughly 40 s and contains ~17 500 frames. The pendulum is launched by hand and the camera motion-triggered. Lighting is a DC-powered LED floodlight (no 50/60 Hz flicker); the background is matte black to make the markers easier to track.

Ground-truth physics (simple-pendulum form)

With two point masses m₁, m₂ on massless rods of length l₁, l₂ hanging from a fixed pivot in uniform gravity g, and angles θ₁, θ₂ measured from the downward vertical, the kinetic and potential energies are

$$T=\frac{1}{2}{m}_1{l}_1^2{\dot{\theta }}_1^2+\frac{1}{2}{m}_2[l_1^2{\dot{\theta }}_1^2+l_2^2{\dot{\theta }}_2^2+2l_1{l}_2{\dot{\theta }}_1{\dot{\theta }}_2\cos ({\theta }_1-{\theta }_2)]T\; =\; \backslash tfrac\{1\}\{2\}\; m\_1\; l\_1^2\; \backslash dot\backslash theta\_1^2\; +\; \backslash tfrac\{1\}\{2\}\; m\_2\; \backslash bigl[\; l\_1^2\; \backslash dot\backslash theta\_1^2\; +\; l\_2^2\; \backslash dot\backslash theta\_2^2\; +\; 2\; l\_1\; l\_2\; \backslash dot\backslash theta\_1\; \backslash dot\backslash theta\_2\; \backslash cos(\backslash theta\_1\; -\; \backslash theta\_2)\; \backslash bigr]$$

$$V=-m_{1}g{l}_1\cos {\theta }_1-m_{2}g(l_1\cos {\theta }_1+l_2\cos {\theta }_2)V\; =\; -m\_1\; g\; l\_1\; \backslash cos\backslash theta\_1\; -\; m\_2\; g\; \backslash bigl(\; l\_1\; \backslash cos\backslash theta\_1\; +\; l\_2\; \backslash cos\backslash theta\_2\; \backslash bigr)$$

with Lagrangian L = T − V and Hamiltonian H = T + V. Applying the Euler–Lagrange equations and solving the resulting 2 × 2 linear system for θ̈₁, θ̈₂ gives the closed form used by formulas/eom_theta1.py and formulas/eom_theta2.py. These match the Wikipedia "Double pendulum" article and Levien & Tan, Am. J. Phys. 61 (1993) 1038. All three formula modules were verified bit-exact against an RK4-integrated trajectory: max |f(X) − θ̈| was numerically 0.0 on 500 test points, and H is conserved to ~1 × 10⁻⁵ J/kg over a 20 s integration (integrator error only).

Why the real-apparatus fit is imperfect

1. Unreported bob masses. IBM's paper gives arm lengths but not m₁, m₂. The formulas assume m₁ = m₂ = 1 kg; because the equations for θ̈₁ and θ̈₂ depend on mass ratios only through m₂ / (2 m₁ + m₂) and m₁ + m₂, the 1:1 assumption is a systematic bias if the upper arm's extra rod mass contributes meaningfully to m₁. The functional form remains correct.
2. Pivot friction and air drag. Real pendulum dynamics dissipate energy. Fig. 4 of Asseman 2018 shows sequences of 200 time-steps (0.5 s each) in which cos/sin of the arm angles clearly decay in amplitude across the run. H is therefore not a conservation law for the real apparatus — it drifts downward. A symbolic-regression system that recovers the full algebraic form of H and reports a non-flat time series on IBM data is doing exactly what a physics-faithful regressor should do: the drift is data, not noise.
3. Template-tracking error. Sub-pixel tracker noise at ~0.2 px in 5× upscaled coordinates translates to an arm-angle noise of ~(0.2 / (l × 5 × px_per_m)) rad per frame. With the centred second-difference estimator of θ̈ this noise is amplified by O(1/dt²) = 1.6 × 10⁵ — so theta{1,2}_ddot_fd are the noise-dominated channels in data.csv, and reference scores on eom_theta1 / eom_theta2 will be bounded from below by tracker noise, not by model mismatch. The hamiltonian target uses first differences only (O(1/dt)) and is much cleaner.
4. Finite-difference scheme. Centred differences are the default here because Asseman 2018 does not prescribe a derivative estimator. Coarser (one-sided) or smoother (Savitzky–Golay) schemes change the scores by order-of-magnitude factors and are documented in data/README.md as the right place to experiment. We deliberately did not apply any filter, so the raw noise floor is visible to the searcher.
5. Launch-by-hand initial conditions. Each of the 21 runs has an unknown initial angular velocity; the 21 trajectories are chaotic and cannot be stitched into a single coherent phase-space orbit. We ship a single video as data.csv by default (contiguous trajectory, ~17 500 rows) and the full concatenation as data_all_videos.csv for completeness.

Column layout

data/data.csv has 16 columns (column 0 = output, columns 1..15 = inputs). See data/README.md for the full schema; the short version is that column 0 is theta1_ddot_fd (output consumed by ground_truth[eom_theta1]), and the two other ground-truth targets (theta2_ddot_fd, H_mpernorm) are included as columns 5 and 6 for row-alignment.

Upstream columns (x_red, y_red, x_green, y_green, x_blue, y_blue, all in 5×-upscaled pixels) are preserved unchanged as columns 7..12. These are not consumed by any current ground-truth formula; they are available for searchers that want to regress angles from pixel positions instead of using the derived theta1, theta2.

Data availability caveat

At the time this entry was authored, the IBM CDN host dax-cdn.cdn.appdomain.cloud was unreachable from the environment. The data/download.sh script is documented and correct, but no raw data.csv has been generated, so metadata.yaml carries reference_scores: null for every ground-truth entry. When the script is re-run on a network with CDN access (or after placing the tarball in data/ by hand), the __main__ block at the bottom of each formulas/*.py will compute the scores against the full 17 500-row single-video data.csv and the 210 000-row data_all_videos.csv.

Contamination tier

high. The two-point-mass double-pendulum Lagrangian, Hamiltonian and Euler–Lagrange equations of motion are in every graduate-level classical mechanics textbook (Goldstein, Taylor, Landau–Lifshitz), the Wikipedia article, hundreds of lecture notes and uncountable Python tutorials. A modern LLM will have memorised the algebraic form verbatim. However:

• The specific apparatus constants (l₁ = 91 mm, l₂ = 70 mm) and the specific data (21 tracked videos from IBM's lab) have much lower memorisation risk — an LLM cannot recite l₁ = 91 mm without looking up Fig. 2b of Asseman 2018.
• The real dissipation (pivot friction + air drag) is not in any textbook because it depends on this particular bearing and these 19 mm disks. A searcher that reports H = const will score badly on this dataset; one that reports H = H₀ exp(-γ t) or similar is doing honest science.

This entry is therefore best used as a Track C (red-team) probe for whether a searcher reproduces the Lagrangian from its own training set, and simultaneously as a Track B (real + scoreable) test of whether it can recover the right dissipation structure. Report results on this dataset alongside results on genuinely low-contamination entries.

Known limitations

• Bob masses not in the paper. All three formulas use m₁ = m₂ = 1 kg; a searcher that recovers any ratio m₁ : m₂ is making a legitimate claim that must be evaluated qualitatively (the dataset cannot pin the masses down without a separate calibration measurement). The overall scale of H is similarly unpinned.
• No ground-truth dissipation model. Asseman 2018 neither measures nor models the pivot friction. A reasonable extension to this entry would be a dissipation ground-truth with form dH/dt = -α ω₁² - β ω₂² and self-fitted coefficients; we have not written this formula here because no reliable source prescribes its form.
• Single default video. data.csv is video 0 out of 21; ONLY_VIDEO=k (0 ≤ k ≤ 20) picks another. data_all_videos.csv contains the full concatenation but is ~210 k rows and should not be used as the primary benchmark data without explicit opt-in.
• Finite-difference noise floor. See §Why the real-apparatus fit is imperfect, point 3.
• License CDLA-Sharing-1.0. Any re-publication of the data must carry the same licence and attribute Asseman, Kornuta & Ozcan (IBM Research AI,
  2018. — the tarball, every derivative data.csv, and this benchmark entry included.