Effects of complex time on periodic and nonperiodic classical trajectories of one-dimensional Hamiltonian systems
Bibliographic record
Abstract
In recent years, much research has been carried out on extending both quantum mechanics and classical mechanics into the complex domain by making parameters of real hermitian Hamiltonians or total energy of the system complex. In this paper we investigate the effects of complex time on periodic and nonperiodic trajectories of both hermitian and nonhermitian one-dimensional classical Hamiltonian systems. Most of the periodic classical trajectories of real hermitian systems turn into nonperiodic and open when the energy or the parameters of the potential become complex. We show that when time is taken as a complex quantity with a specific fixed phase angle or as a specific complex function, nonperiodic trajectories become periodic and closed. Furthermore, we show that real hermitian systems, such as H = p2/2m + x4 + bx3 + cx2 + dx (b, c, and d are real quantities) possess classical periodic trajectories for real energies even when time is complex (i.e., t = treiτ). It was found that there is a discrete set of τ values for which the trajectories of the preceding system are closed and periodic and periods associated with them form a countably infinite set.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".