PreprintCC BY 4.0arXiv
A generalization of Gordon's Theorem to arbitrary winding number
Abstract
We study minimizers of the Keplerian action functional with homogeneous potential of type 1/|u|α on periodic loops with fixed winding number k ∈ ℤ∖{0}. This extends the classical theorems of Gordon and Venturelli, closing the previously open case α∈ (1,2), |k|≥ 2. We show that the k-circular solution loses local minimality below the threshold α= 2−1/|k|2, and that minimizers are collision-free above α= 2−1/|k|. Combining these results with the existence of precessing periodic orbits in between, we obtain a full classification of the minimizer in terms of α and k, which we also verify numerically for k=2.
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BibTeX
@misc{fenucci2026generalization,
title = {{A generalization of Gordon's Theorem to arbitrary winding number}},
author = {M. Fenucci},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31872},
eprint = {2610.31872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31872},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A generalization of Gordon's Theorem to arbitrary winding number
AU - M. Fenucci
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31872
UR - https://arxiv.org/abs/2610.31872
AB - We study minimizers of the Keplerian action functional with homogeneous potential of type 1/|u|^α on periodic loops with fixed winding number k ∈ ℤ∖{0}. This extends the classical theorems of Gordon and Venturelli, closing the previously open case α∈ (1,2), |k|≥ 2. We show that the k-circular solution loses local minimality below the threshold α= 2−1/|k|^2, and that minimizers are collision-free above α= 2−1/|k|. Combining these results with the existence of precessing periodic orbits in between, we obtain a full classification of the minimizer in terms of α and k, which we also verify numerically for k=2.
ER -