Regular and irregular motion: new mechanical results and fibre optics analogies

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Abstract
The authors present some new results concerning regular and irregular motions in classical systems. The origin and motivation for the work is to be found in fibre optics. They show that the equations describing ray paths in axially uniform optical fibres are mathematically equivalent to the classical mechanics of a particle moving in a two-dimensional potential U. The classification of rays in non-circular cross-section fibres is related to questions of integrability and the existence of regular motion in mechanical systems. The elliptic fibre leads to potentials U(w) where w2=x2+A2y2 where A is a constant. It is conjectured that such potentials give rise to regular motion as revealed by a study of phase space trajectories, but the second invariant is unknown. The special case U=wq is studied in detail. These potentials belong to the wider class in which U is a homogeneous function of x and y, but not all potentials in that class give rise to regular motions.