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

2025/06/30 by Ladeishchikov, E. A., Lokutsievskiy, L. V., Prilepin, N. V.
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2507.00250

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_Ω and \sinh_Ω, developed here, prove especially useful; they generalize the classical \cosh and \sinh to the case of an unbounded convex set Ω⊂ℝ².

Citations

• Lorentzian distance on the Lobachevsky plane
• Antinorms on cones: duality and applications
• Antinorms on cones: duality and applications
• Explicit formulae for geodesics in left invariant sub-Finsler problems on Heisenberg groups via convex trigonometry
• Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry
• Sub-Riemannian and sub-Lorentzian geometry on \SU(1,1) and on its universal cover
• Sub-Lorentzian Geometry on Anti-De Sitter Space

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