The Lagrangian and symplectic structures of the Kuramoto oscillator model

Summary

The document explores the Lagrangian and symplectic structures of the Kuramoto oscillator model, a key example of non-equilibrium phase transitions. It demonstrates that despite the original equations lacking a Lagrangian form, a variational description can be derived by considering the Kuramoto model as a mean-field classical spin model. The study also establishes connections between Kuramoto synchronization and the spin pairing mechanisms found in the semiclassical Gaudin model.

Keywords

• Kuramoto model

• non-equilibrium phase transitions

• synchronization

• Lagrangian structure

• Heisenberg spin models

• semiclassical Gaudin model

Main claims

• The Kuramoto oscillator model can be variationally described as a mean-field classical twisted spin model on S2.

• Perturbation analysis around unstable Kuramoto equilibria is equivalent to low-energy fluctuations of mean-field Heisenberg spin models.

• Off-plane perturbations around Kuramoto equilibria are described by a semiclassical Gaudin model, linking oscillator synchronization to spin pairing mechanisms.