Geometric phase of a classical Aharonov-Bohm Hamiltonian

We present a gauge-invariant approach for associating a geometric phase with the phase space trajectory of a classical dynamical system. As an application, we consider the classical analog of the quantum Aharonov-Bohm (AB) Hamiltonian for a charged particle orbiting around a current carrying long thin solenoid. We compute the classical geometric phase of a closed phase space trajectory, and also determine its dependence on the magnetic flux enclosed by the orbit. We study the similarities and differences between this classical geometric phase and the AB phase acquired by the wave function of the quantum AB Hamiltonian. We suggest an experiment to measure the geometric phase for the classical AB system, by using an appropriate optical fiber ring interferometer.