Total collision in a four-body problem with Jacobi potential
Bibliographic record
Abstract
Abstract We study the dynamics of a spatial four-body problem where the bodies maintain a rhombus-shape configuration at all times: two of the bodies of equal mass move in the horizontal plane symmetrically with respect to the origin while another pair of bodies of equal mass move symmetrically opposed along the vertical axis. The bodies interact via the Jacobi potential, an attractive binary potential of the form $-1/x^2$, where $x$ is the distance between the particles. We use appropriate transformations to blow up total collision into a manifold pasted onto the phase space for all levels of energy. We find that the topology of the total collision manifold changes as the angular momentum varies, and also that the dynamics on the collision manifold changes as a function of angular momentum and the mass ratio. We also give a qualitative description of the global flow of the problem for negative energy. This description utilizes knowledge concerning the flow on and near the collision manifold, the presence of an additional integral of motion, and takes advantage of the time-reversing symmetry inherent in the system of equations.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".