Collisional dynamics for strong-weak potential Hill's lunar problem
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Bibliographic record
Abstract
We study a particular dynamical system in terms of global existence and singularity. This model has various physical applications. For instance, it can be derived from the quasi-homogeneous three-body system in a rotating coordinate system with angular speed $ \omega $. On the other hand, quasi-homogeneous ($ U(r) = -\frac{A}{r^a}-\frac{B}{r^b} $, where $ r $ is the mutual distance and $ A,B,a,b $ are positive constants) potential itself is of great interest. Important examples of quasi-homogeneous potentials include the Schwarzschild potential ($ U_{\text{Schwarzschild}}(r) = -\frac{A}{r}-\frac{B}{r^3} $) and the Manev potential ($ U_{\text{Manev}}(r) = -\frac{A}{r}-\frac{B}{r^2} $).This paper is partitioned into two major parts. In the first part, we classify the fate of a given initial condition dynamically under the flow of the dynamical system. We fully characterize the phase space under some energy threshold using an indicator function. This energy threshold is characterized variationally. In particular, for strong-weak potentials ($ b>2>a $), our results demonstrate the existence of the 'black hole effect' for $ \omega $ sufficiently large: Collision sets, with non-zero Lebesgue measure, consisting of initial conditions that lead to finite-time collision, are constructed under and at an energy threshold.In the second part, we introduce the McGehee coordinate system, as well as a new time scale, to study near-collision dynamics of the system. By applying the McGehee transformation, we are able to derive the asymptotic configuration with respect to time for any collision solution at collision.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 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 it