# 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

$$T = \tfrac{1}{2} m_1 l_1^2 \dot\theta_1^2
      + \tfrac{1}{2} m_2 \bigl[ l_1^2 \dot\theta_1^2 + l_2^2 \dot\theta_2^2
        + 2 l_1 l_2 \dot\theta_1 \dot\theta_2 \cos(\theta_1 - \theta_2) \bigr]$$

$$V = -m_1 g l_1 \cos\theta_1 - m_2 g \bigl( l_1 \cos\theta_1 + l_2 \cos\theta_2 \bigr)$$

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.