On Kostant-Kirillov Symplectic Structure and Quasi-Poisson Structures of the Euler-Arnold Systems
Bibliographic record
Abstract
The concept of symplectic structure emerged between 1808 to 1810 through the works of Lagrange and Poisson on the trajectory of the planets of the solar system. In order to explain the variation of the orbital parameters, they introduced the symplectic structure associated to the manifold describing the states of the system and a fundamental operation on functions called Poisson’s bracket. But, the latter also comes from the Hamiltonian formalism which does not automatically lead to a Poisson structure. Although contrary to the Riemannian case, not every manifold necessarily admits a symplectic structure including even dimensional manifolds. The aim of this paper is to show the interaction between the Kostant-Kirillov symplectic structure and quasi-Poisson structures coming from the Euler-Arnold systems. The Lie algebra theoretical approach based on the Kostant-Kirillov coadjoint action will allow us to obtain a class of the quasi-Poisson structures resulting from the characterization of the Hamiltonian system and to prove some results on the Kostant-Kirillov symplectic structure in the quasi-Poisson context.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".