Bibliographic record
Abstract
The concept of integrability of Hamiltonian systems goes back at least to the 19th century. The idea of integrability in classical mechanics was formalised by J. Liouville. A finite-dimensional Hamiltonian system with n degrees of freedom is called “Liouville integrable” or “completely integrable” if it allows n functionally independent integrals of motion that are well defined functions on phase space and are in involution. In classical mechanics the equations of motion for a Liouville integrable system can be, at least in principle, reduced to quadratures. A completely integrable system in quantum mechanics is defined similarly. It should allow n commuting integrals of motion (including the Hamiltonian) that are well defined operators in the enveloping algebra of the Heisenberg algebra, or some generalization of this enveloping algebra. In quantum mechanics complete integrability does not guarantee that the spectral problem for the Schrödinger operator can be solved explicitly, or even that the energy levels can be calculated algebraically. An n -dimensional integrable Hamiltonian system that admits more than n integrals of motion is called “superintegrable”. Systems with 2 n – 1 integrals, with at least one subset of n of them in involution, are “maximally superintegrable”. Such systems, namely the Kepler-Coulomb system and the harmonic oscillator, played a pivotal role in the development of physics and mathematics. Trajectories in classical maximally superintegrable systems can at least in principle be calculated algebraically (without using any calculus). Ironically, calculus was invented in order to calculate orbits in the Kepler system.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.216 | 0.096 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".