Bibliographic record
Abstract
Using geometric mechanics methods, we examine aspects of the dynamics of n mass points in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msup></mml:math> with a general pairwise potential. We investigate the central force problem, set up the n -body problem and discuss certain properties of relative equilibria. We describe regular n -gons in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msup></mml:math> , and when the masses are equal we determine the invariant manifold of motions with regular n -gon configurations. In the case n = 3, we reduce the dynamics to a 6 d.f. system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration are unstable. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".