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We study the concepts of adiabatic driving and geometric phases of classicalintegrable systems under the Koopman-von Neumann formalism. In close relationto what happens to a quantum state, a classical Koopman-von Neumann eigenstatewill acquire a geometric phase factor $exp\left\{ i\Phi\right\} $ after aclosed variation of the parameters $\lambda$ in its associated Hamiltonian. Theexplicit form of $\Phi$ is then derived for integrable systems, and itsrelation with the Hannay angles is shown. Additionally, we use quantum formulasto write a classical adiabatic gauge potential that generates adiabatic unitaryflow between classical eigenstates, and we explicitly show the relationshipbetween the potential and the classical geometric phase.

Adiabatic driving and geometric phases in classical systems