Störmer problem restricted to a spherical surface
Why this work is in the frame
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Bibliographic record
Abstract
In order to analyse in full detail the dynamics of a charged particle in the field of a magnetic dipole, we propose to study the restricted motion of the particle in a spherical surface with the dipole at its centre. This model can be considered as the classical non-relativistic Störmer problem within a sphere, and although this problem no longer represents the real Störmer problem, it shows the complex behaviour of this magnetic field through the classical dynamics equations that can be formally integrated. We start from a Lagrangian approach which allows us to analyse the dynamical properties of the system, such as the role of a velocity dependent potential, the symmetries and the conservation properties. We derive the Hamilton equations of motion, which in this restricted case can be reduced to a quadrature. From the Hamiltonian function we find, for the polar angle, an equivalent one-dimensional system of a particle in the presence of an effective potential. This equivalent potential function, which is a double well potential, allows us to get a clear description of the dynamics of the system. Then we obtain, by means of numerical integration, different plots of the trajectories in three-dimensional graphs in the sphere. This restricted case of the Störmer problem is still nonlinear, with complex and interesting dynamics and we believe that it can offer the student a better grasp of the subject than the general three-dimensional case.
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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) | 0.000 | 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 it