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

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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.

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Notes

  1. This turned out to be the anti-symmetric partner of “deterministic friction”, a dissipative force proposed by Wilkinson [6].

  2. An informal way to define ergodicity is that the system finishes by being more or less everywhere in phase space if one waits long enough.

  3. The definition of a generic f is given in [4], \(<f>_{E}=(1/\partial _{E}{\Omega } )\int ~dz~\delta (E-h)f(z,\mathbf {R})\) where the normalization is the phase space volume on the energy shell \(\partial _{E}{\Omega } =\int ~dz~\delta (E-h)\) satisfying h(z, R) = E. The total phase space volume inside the E-shell \({\Omega } (E)=\int ~dz~\delta (E-h)\) is independent of R.

  4. I thank Professor Berry for pointing this out.

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Acknowledgments

It is a pleasure to thank Professor John Hannay for many helpful correspondences and concrete suggestions. Also I am grateful to Professor Michael Berry for comments. Also I thank the referee for the constructive comments and suggestions.

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This work was not supported by any funding agency.

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  1. Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, 700108, India

    Subir Ghosh

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Ghosh, S. Geometric Phases for Classical and Quantum Dynamics: Hannay Angle and Berry Phase for Loops on a Torus. Int J Theor Phys 58, 2859–2871 (2019). https://doi.org/10.1007/s10773-019-04169-6

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