Controllability of a system of two symmetric rigid bodies in three space
Bibliographic record
Abstract
Imposing the constraint that a mechanical system having two completely symmetric 3-D rigid bodies have zero total angular momentum, the angular velocities of these bodies are negatives of one another, and the transfer of this sytem from one position to another is nonholonomic. While Chow's theorem establishes the fixed-endpoint controllability of the system, this result does not explicitly exhibit any motions between a given set of endpoints. In the present work it is shown how to explicitly construct simple motions of this system on (0, 1) with arbitrary endpoints in SO(3)/sup 2/. In particular, using normalized quaternions to describe rotations, the author constructs, in terms of elementary functions, a continuous motion with the given endpoints; it consists of at most three successive motions during each of which the bodies rotate on fixed axes. It is assumed that the rigid bodies have coincident mass centers, that each has a completely symmetric mass distribution about its mass center, and that the inertia tensor of each body is the identity matrix.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
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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".