Particle on a Torus Knot: Anholonomy and Hannay Angle

Abstract

The phenomenon of rotation of a vector under parallel transport along a closed path is known as anholonomy. In this paper we have studied the anholonomy for noncontractible loops - closed paths in a curved surface that do not enclose any area and hence Stokes theorem is not directly applicable. Examples of such closed paths are poloidal and toroidal loops and knots on a torus. The present study is distinct from conventional results on anholonomy for closed paths on $S_2$ since in the latter case all closed paths are contractible or trivial cycles. We find that for some nontrivial cycles the anholonomy cancels out over the complete cycle. Next we calculate Hannay angle for a particle traversing such noncontractible loops when the torus itself is revolving. Some new and interesting results are obtained especially for poloidal paths that is for paths that encircle the torus ring.

Results of first part of original paper (anholonomy and Hannay angle), paper rewritten, title changed, accepted in International Journal of Geometric Methods in Modern Physics (IJGMMP)

References in corpus (5)

• Quantum field theory on toroidal topology: algebraic structure and applications
• The Classical and Quantum Mechanics of a Particle on a Knot
• Particle on a Torus Knot: Constrained Dynamics and Semi-Classical Quantization in a Magnetic Field
• The Two Dimensional Hannay-Berry Model
• The Multidimensional Berry-Hannay Model

Cited by in corpus (2)

• Fermions on a torus knot
• Geometric Phases for Classical and Quantum Dynamics: Hannay angle and Berry Phase for Loops on a Torus