Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups

Abstract

In this paper, we consider the problem of finding geodesics in a series of left-invariant problems endowed with sub-Lorentzian and Finsler structures. Explicit formulas for extremals are obtained in terms of convex trigonometric functions. In the sub-Lorentzian setting, the new trigonometric functions ${\cosh }_{\Omega }$ and ${\sinh }_{\Omega }$, developed here, prove especially useful; they generalize the classical $\cosh$ and $\sinh$ to the case of an unbounded convex set $\Omega \subset {\mathbb{R}}^2$.