Geodesic
Extremal trajectories in the sub-Lorentzian problem on the Engel group
Sbornik. Mathematics, Tome 209 (2018) no. 11, pp. 1547-1574

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Let $\mathbb{E}$ be the Engel group and let $D$ be a rank-two left-invariant distribution with Lorentzian metric on $\mathbb{E}$. The sub-Lorentzian problem is stated as the problem of maximizing the sub-Lorentzian distance. A parametrization of timelike and spacelike normal extremal trajectories is obtained in terms of Jacobi elliptic functions. Discrete symmetry groups are described in the cases of timelike and spacelike trajectories; in both cases the fixed points and the corresponding Maxwell points are calculated for each symmetry. These calculations underlie estimates for the cut time (when the trajectory ceases to be globally optimal). Bibliography: 17 titles.
Keywords: Engel group, extremal trajectories, sub-Lorentzian metric, Jacobi functions. 
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     title = {Extremal trajectories in the {sub-Lorentzian} problem on the {Engel} group},
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A. A. Ardentov; Yu. L. Sachkov; T. Huang; X. Yang. Extremal trajectories in the sub-Lorentzian problem on the Engel group. Sbornik. Mathematics, Tome 209 (2018) no. 11, pp. 1547-1574. http://geodesic.mathdoc.fr/item/SM_2018_209_11_a0/