Formal Integrals of Motion in Time Periodic Hamiltonian Systems
Bibliographic record
Abstract
We present an algorithm for the construction of approximate integrals of motion in time periodic Hamiltonian systems of the form H=H₀+ε H₁ where H₀=(ω₁²x²+y²)/2. We apply this algorithm in the case where H₁=−ε x³cos(ω t) with ω/ω₁ irrational and calculate approximate integrals of motion Φ=Φ₀+εΦ₁+... close to the stable periodic orbits of the system. These integrals agree approximately with the form the stroboscopic sections found by the numerical integration of the system. The agreement is better for smaller values of ε. In the resonant cases where ω/ω₁ is rational we have secular terms (terms proportional to t) in some Φᵢ. These secular terms may be avoided by using a combination of Φ and another formal integral with more complicated zero order term. We calculated explicitly such integrals in the case H₁=−ε x³ cos(ω t) the resonances with ω=2ω₁, ω=3ω₁ and in the case H₁=−ε x² cos(ω t) (in which the dynamics is governed by a Mathieu equation) the resonance ω=ω₁.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".