Lagrangian and Hamiltonian Formulation of Classical Electrodynamics\n without Potentials
Bibliographic record
Abstract
In the standard Lagrangian and Hamiltonian approach to Maxwell's theory the\npotentials $A^{\\mu}$ are taken as the dynamical variables. In this paper I take\nthe electric field $\\vec{E}$ and the magnetic field $\\vec{B}$ as the the\ndynamical variables. I find a Lagrangian that gives the dynamical Maxwell\nequations and include the constraint equations by using Lagrange multipliers.\nIn passing to the Hamiltonian one finds that the canonical momenta\n$\\vec{\\Pi}_E$ and $\\vec{\\Pi}_B$ are constrained giving 6 second class\nconstraints at each point in space. Gauss's law and\n$\\vec{\\nabla}\\cdot\\vec{B}=0$ can than be added in as additional constraints.\nThere are now 8 second class constraints, leaving 4 phase space degrees of\nfreedom. The Dirac bracket is then introduced and is calculated for the field\nvariables and their conjugate momenta.\n
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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 | 0.000 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 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".