  The Kuramoto model constitutes a paradigmatic model for the dissipative
collective dynamics of coupled oscillators, characterizing in particular the
emergence of synchrony. Here we present a classical Hamiltonian (and thus
conservative) system with 2N state variables that in its action-angle
representation exactly yields Kuramoto dynamics on N-dimensional invariant
manifolds. We show that the synchronization transition on a Kuramoto manifold
emerges where the transverse Hamiltonian action dynamics becomes unstable. The
uncovered Kuramoto dynamics in Hamiltonian systems thus distinctly links
dissipative to conservative dynamics.
