Lagrangian and Hamiltonian Formulation of Classical Electrodynamics\n without Potentials
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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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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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 it