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