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Sub-Lorentzian distance and spheres on the Heisenberg group

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Abstract

The left-invariant sub-Lorentzian problem on the Heisenberg group is considered. An optimal synthesis is constructed, the sub-Lorentzian distance and spheres are described.

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Notes

  1. where \(\pi \, : \, T^{*}M \rightarrow {M}\) is the canonical projection, \(\pi (\lambda ) = q, \lambda \in T^{*}_{q}M\)

  2. where \(\vec {h}(\lambda )\) is the Hamiltonian vector field on \(T^*M\) with the Hamiltonian function \(h(\lambda )\)

  3. The set \(\{(h_1, h_2) \in (\mathbb {R}^2)^* \mid h_1 \le - |h_2|\}\) is the polar set to U in the sense of convex analysis.

  4. The symplectic foliation is the decomposition of the dual of a Lie algebra into coadjoint orbits, see e.g. [2]

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  1. Ailamazyan Program Systems Institute of RAS, Pereslavl-Zalessky, Russia

    Yuri L. Sachkov & Elena F. Sachkova

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  1. Yuri L. Sachkov
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Correspondence to Yuri L. Sachkov.

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Work supported by Russian Scientific Foundation, grant 22-11-00140, https://rscf.ru/project/22-11-00140/.

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Sachkov, Y.L., Sachkova, E.F. Sub-Lorentzian distance and spheres on the Heisenberg group. J Dyn Control Syst 29, 1129–1159 (2023). https://doi.org/10.1007/s10883-023-09652-2

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