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

TL;DR

This paper analyzes extremal trajectories in a sub-Riemannian problem on the pseudo Euclidean motion group SH(2), deriving explicit formulas using elliptic functions and characterizing Maxwell strata through symmetry analysis.

Contribution

It provides a detailed integration of the sub-Riemannian geodesics on SH(2) using elliptic functions and characterizes Maxwell strata via symmetry considerations.

Findings

Extremal trajectories are expressed with Jacobi elliptic functions.

Projections of trajectories have cusps and inflection points.

Maxwell strata are characterized using reflection symmetries.

Abstract

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.