Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
Bibliographic record
Abstract
We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti.This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor.First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors.This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates.Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation.As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space.We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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".