General Principles of Hamiltonian Formulations of the Metric Gravity
Bibliographic record
Abstract
Principles of successful Hamiltonian approaches, which were developed to describe free gravitational field(s) in the metric gravity, are formulated and discussed. By using the standard $$\Gamma{-}\Gamma$$ Lagrangian $${\mathcal{L}}_{\Gamma{-}\Gamma}$$ of the metric GR we properly introduce all momenta of the metric gravitational field and derive the both canonical $$H_{C}$$ and total $$H_{t}$$ Hamiltonians of the metric GR. We also developed an effective method which is used to determine various Poisson brackets between analytical functions of the basic dynamical variables, i.e., generalized coordinates $$g_{\alpha\beta}$$ and momenta $$\pi^{\mu\nu}$$ . In general, such variables can be chosen either from the straight $$\{g_{\alpha\beta},\pi^{\mu\nu}\}$$ , or dual $$\{g^{\alpha\beta},\pi_{\mu\nu}\}$$ sets of symplectic dynamical variables which always arise (and complete each other) in any Hamiltonian formulation developed for the coupled system of tensor fields. By applying canonical transformation(s) of dynamical variables we reduce the canonical Hamiltonian $$H_{C}$$ to its natural form. The natural form of canonical Hamiltonian provides numerous advantages in actual applications to the metric GR, since the general theory of dynamical systems with such Hamiltonians is well developed. Furthermore, many analytical and numerically exact solutions have been found and described in detail for dynamical systems with the Hamiltonians already reduced to their natural forms. In particular, reduction of the canonical Hamiltonian $$H_{C}$$ to its natural form allows one to derive the Jacobi equation for the free gravitational field(s), which takes a particularly simple form.
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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 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| 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".