Multiplicity of Finsler geodesics and electromagnetic trajectories via localized deformations
Abstract
Given a Finsler manifold (M,F) and suitable nested regions B1⊂ B2⊂ B3, with M∖ B1 connected and non-contractible, as in the model case of concentric balls in ℝn, n≥2, we consider conformal deformations φλF which can be chosen so as to leave F unchanged outside B3 and grow on B2∖ B1. This growth creates an energy barrier, diverging with λ, that low-energy curves cannot cross. The minimax families can therefore be chosen in the path space of M∖ B1, which contains compact subsets of arbitrarily large category. Consequently, for every m∈ℕ and all sufficiently large λ, there exist at least m geodesics joining two prescribed points, all avoiding B1. This extends to the Finsler setting an idea of Capozzi, Fortunato and Greco. The same scheme applies to Randers metrics α+β, with α deformed and β unchanged, as well as to fixed-energy trajectories of non-relativistic electromagnetic systems and relativistic Lorentz-force trajectories, and light rays in spacetimes endowed with a causal Killing field. In the non-relativistic case only the scalar potential is perturbed, while the magnetic field is preserved. When the Killing field is not everywhere timelike, the Fermat principle yields a possibly singular Randers--Kropina metric on a spacelike hypersurface. Under a natural admissible category condition, combining localized barriers with Randers approximation gives m Randers--Kropina geodesics with pairwise distinct lengths, equivalently m future-pointing light rays with pairwise distinct arrival times.
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BibTeX
@misc{caponio2026multiplicity,
title = {{Multiplicity of Finsler geodesics and electromagnetic trajectories via localized deformations}},
author = {Erasmo Caponio and Anna Valeria Germinario and Antonio Masiello},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03533},
eprint = {2610.03533},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03533},
}
RIS
TY - UNPB
TI - Multiplicity of Finsler geodesics and electromagnetic trajectories via localized deformations
AU - Erasmo Caponio
AU - Anna Valeria Germinario
AU - Antonio Masiello
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03533
UR - https://arxiv.org/abs/2610.03533
AB - Given a Finsler manifold (M,F) and suitable nested regions B_1⊂ B_2⊂ B_3, with M∖ B_1 connected and non-contractible, as in the model case of concentric balls in ℝ^n, n≥2, we consider conformal deformations φ_λF which can be chosen so as to leave F unchanged outside B_3 and grow on B_2∖ B_1. This growth creates an energy barrier, diverging with λ, that low-energy curves cannot cross. The minimax families can therefore be chosen in the path space of M∖ B_1, which contains compact subsets of arbitrarily large category. Consequently, for every m∈ℕ and all sufficiently large λ, there exist at least m geodesics joining two prescribed points, all avoiding B_1. This extends to the Finsler setting an idea of Capozzi, Fortunato and Greco. The same scheme applies to Randers metrics α+β, with α deformed and β unchanged, as well as to fixed-energy trajectories of non-relativistic electromagnetic systems and relativistic Lorentz-force trajectories, and light rays in spacetimes endowed with a causal Killing field. In the non-relativistic case only the scalar potential is perturbed, while the magnetic field is preserved. When the Killing field is not everywhere timelike, the Fermat principle yields a possibly singular Randers--Kropina metric on a spacelike hypersurface. Under a natural admissible category condition, combining localized barriers with Randers approximation gives m Randers--Kropina geodesics with pairwise distinct lengths, equivalently m future-pointing light rays with pairwise distinct arrival times.
ER -