The Foucault pendulum viewed as a spherical pendulum placed in a rotating reference system
Bibliographic record
Abstract
For a pendulum of a given length, installed at any fixed latitude, the motion depends on three variables: the total mechanical energy, connected with the maximum angular distance of the bob from the vertical; the projection of the angular momentum along the vertical, connected with the shape of its orbit along the surface of a sphere, which may be described as a veering ellipse recalling the orbit of Mercury; and the angular velocity, Ω, of the rotating reference system given by that of the Earth. When the value of one of these variables is negligible, the ensuing motion has already been calculated in analytical grounds: a plummet, a Foucault pendulum, or a spherical pendulum including as a special case the so-called plane pendulum. This paper offers a Lagrangian approach in the case when Ω is the sidereal angular velocity of the Earth. To our knowledge, these are the first analytical solutions for any boundary conditions. When the boundary conditions are such that the bob passes through the vertical (i.e., the direction defined by the plummet), the behavior of the pendulum is as expected: the veering velocity of the plane equals the angular velocity of the Earth times the sine of the latitude. This state is reached when the projection of its angular momentum along the vertical vanishes. However, if that projection takes any other value, the motion is produced along a veering ellipse and the time rate of advance of the major axis is the sum of the previous result plus the spherical pendulum contribution. Our solutions include, as special cases, those of the spherical pendulum, or the ideal Foucault pendulum with finite amplitude.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".