# Geometric Phases for Classical and Quantum Dynamics: Hannay angle and Berry Phase for Loops on a Torus

## TL;DR

This paper investigates the geometric phases, specifically Hannay angle and Berry phase, for particles moving along noncontractible loops on a torus, considering the effects of adiabatic revolutions of the torus.

## Contribution

It introduces a computational scheme that calculates geometric phases for noncontractible loops on a torus without relying on Stokes theorem.

## Key findings

- Calculated Hannay angle and Berry phase for various loops on a torus.
- Developed a line integral based method avoiding Stokes theorem complications.
- Analyzed the effects of adiabatic torus revolutions on geometric phases.

## Abstract

In this paper we have considered closed trajectories of a particle on a two-torus where the loops are noncontractible (poloidal and toroidal loops and knots embedded on a regular torus). We have calculated Hannay angle and Berry phase for particle traversing such loops and knots when the torus itself is adiabatically revolving. Since noncontractible loops do not enclose any area Stokes theorem has to be applied with caution. In our computational scheme we have worked with line integrals directly thus avoiding Stokes theorem.