Instability of Periodic Orbits of Some Rhombus and Parallelogram Four Body Problems
Bibliographic record
Abstract
The rhombus and parallelogram orbits are interesting families of periodic solutions, which come from celestial mechanics and the N -body problem.Variational methods with finite order symmetry group are used to construct minimizing non-collision periodic orbits.We study the question of stability or instability of periodic and symmetric periodic solutions of the rhombus and the equal mass parallelogram four body problems.We first study the stability of periodic solutions for the rhombus four body problem.An analytical description of the variational principle is used to show that the homographic solutions are the minimizers of the action functional restricted to rhombus loop space [23].We employ techniques from symplectic geometry and specifically a variant of the Maslov index for curves of Lagrangian subspaces along the minimizing rhombus orbit to prove the main result, Theorem 4.2.2, which states that the reduced rhombus orbit is hyperbolic in the reduced energy manifold when it is not degenerate.We second study the stability for symmetric periodic solutions of the equal mass parallelogram four body problem.The parallelogram family is a family of Z 2 Z 4 symmetric action minimizing solutions, investigated by [7].In this example, the minimizing solution [7] can be extended to a 4T -periodic solution using symmetries through square and collinear configurations.The Maslov index of the orbits is used i to prove the main result, Theorem 5.3.1, which states that the minimizing equal mass parallelogram solution is unstable when it is non-degenerate.
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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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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