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We consider the sub-Riemannian length minimization problem on the group ofmotions of pseudo Euclidean plane that form the special hyperbolic group SH(2).The system comprises of left invariant vector fields with 2-dimensional linearcontrol input and energy cost functional. We apply the Pontryagin MaximumPrinciple to obtain the extremal control input and the sub-Riemanniangeodesics. A change of coordinates transforms the vertical subsystem of thenormal Hamiltonian system into the mathematical pendulum. In suitable ellipticcoordinates the vertical and the horizontal subsystems are integrated such thatthe resulting extremal trajectories are parametrized by the Jacobi ellipticfunctions. Qualitative analysis reveals that the projections of normal extremaltrajectories on the xy-plane have cusps and inflection points. The verticalsubsystem being a generalized pendulum admits reflection symmetries that areused to obtain a characterization of the Maxwell strata.

Extremal Trajectories and Maxwell Strata in Sub-Riemannian Problem on Group of Motions of Pseudo Euclidean Plane