Explicit formulae for extremals in sub-Lorentzian and Finsler problems on 2D and 3D Lie groups

E. A. Ladeishchikov^a, L. V. Lokutsievskiy^b, N. V. Prilepin<sup>a

a</sup> Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: Questions relating to the search of geodesics in a series of left-invariant problems with sub-Lorentzian or Finsler structure are under consideration. Explicit formulae for extremals are found in terms of trigonometric functions of convex trigonometry. In sub-Lorentzian problems the machinery of the new trigonometric functions $\sinh_\Omega$ and $\cosh_\Omega$, generalizing $\sinh$ and $\cosh$ to the case of an unbounded convex set $\Omega\subset \mathbb R^2$, is particularly useful.
Bibliography: 18 titles.

Keywords: sub-Lorentzian geometry, sub-Finsler geometry, convex trigonometry, Pontryagin's maximum principle, 3D unimodular Lie groups.

MSC: Primary 49K15, 53B30, 53B40; Secondary 22E30, 33E99

Received: 29.05.2025

DOI: 10.4213/sm10355

 English version: Sbornik: Mathematics, 2025, 216:12, 1713–1753