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Explicit Formulae for Geodesics in Left–Invariant Sub–Finsler Problems on Heisenberg Groups via Convex Trigonometry

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Abstract

In the present paper, we obtain explicit formulae for geodesics in some left–invariant sub–Finsler problems on Heisenberg groups \(\mathbb {H}_{2n+1}\). Our main assumption is the following: the compact convex set of unit velocities at identity admits a generalization of spherical coordinates. This includes convex hulls and sums of coordinate 2–dimensional sets, all left–invariant sub–Riemannian structures on \(\mathbb {H}_{2n+1}\), and unit balls in Lp–metric for \(1\le p\le \infty \). In the last case, extremals are obtained in terms of incomplete Euler integral of the first kind.

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Notes

  1. Obviously, for any 𝜃, there exists an angle 𝜃∘ such that \(\theta ^{\circ }\leftrightarrow {{\varOmega }^{\circ }}\theta \). This can be easily proved by the hyperplane separation theorem.

  2. The angle 𝜃 is defined up to \(2\mathbb {S}\mathbb {Z}\) as always.

  3. I.e., a solution to PMP projection on the base \(\mathbb {H}_{2n+1}\).

  4. Since λi(t) and 𝜃i(t) have a lot of freedom, nothing more can be said about x, y, and z in the general case γ = 0 (this situation is completely similar to the case n = 1, see [5, 16]).

  5. Obviously, hi, gi, and γ are linear on fibers left–invariant coordinates on \(T^{*}\mathbb {H}_{2n+1}\).

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Acknowledgments

The author would like to express his deep gratitude to Professor A.I. Nazarov for interesting discussions and pointing on a connection to Shelupsky’s functions.

Funding

This work is supported by the Russian Science Foundation under grant 20-11-20169.

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  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    L. V. Lokutsievskiy

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  1. L. V. Lokutsievskiy
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Correspondence to L. V. Lokutsievskiy.

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Lokutsievskiy, L.V. Explicit Formulae for Geodesics in Left–Invariant Sub–Finsler Problems on Heisenberg Groups via Convex Trigonometry. J Dyn Control Syst 27, 661–681 (2021). https://doi.org/10.1007/s10883-020-09516-z

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