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PreprintCC BY 4.0arXiv

Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance

  • Lev Lokutsievskiy
  • Alexander Plakhov
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Abstract

We consider the following model in the framework of Newtonian aerodynamics: a 2D convex body moves forward and slowly oscillates in a rarefied medium on the plane. The law of oscillations is given. The problem is to find a body of fixed area that has the smallest resistance. We solve this problem by reducing it to an isoperimetric problem for quasi-Finsler metrics. Further, we introduce two criteria of smallness of oscillations. We show that the optimal body has singularity at the front (back) part of its boundary iff the former (latter) criterion is satisfied. The rest of the boundary is smooth. Finally, we find exact optimal shapes in the case of uniform oscillations and construct several shapes explicitly.

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Cite

BibTeXRIS
BibTeX
@misc{lokutsievskiy2026isoperimetric,
  title = {{Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance}},
  author = {Lev Lokutsievskiy and Alexander Plakhov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.17783},
  eprint = {2608.17783},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.17783},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance
AU  - Lev Lokutsievskiy
AU  - Alexander Plakhov
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.17783
UR  - https://arxiv.org/abs/2608.17783
AB  - We consider the following model in the framework of Newtonian aerodynamics: a 2D convex body moves forward and slowly oscillates in a rarefied medium on the plane. The law of oscillations is given. The problem is to find a body of fixed area that has the smallest resistance. We solve this problem by reducing it to an isoperimetric problem for quasi-Finsler metrics. Further, we introduce two criteria of smallness of oscillations. We show that the optimal body has singularity at the front (back) part of its boundary iff the former (latter) criterion is satisfied. The rest of the boundary is smooth. Finally, we find exact optimal shapes in the case of uniform oscillations and construct several shapes explicitly.
ER  -