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The Lagrangian and symplectic structures of the Kuramoto oscillator model

Sherwin Kouchekian, Razvan Teodorescu

TL;DR

This work establishes a variational framework for the Kuramoto oscillator by embedding it into a mean-field classical spin system on $S^2$, deriving a corresponding Lagrangian and Hamiltonian that capture both planar and off-plane perturbations. Planar perturbations recover known synchronization stability results, while off-planar fluctuations are shown to be governed by a semiclassical Gaudin/Richardson spin-pairing structure, linking oscillator synchronization to spin-pairing physics. The authors further develop a geometric quantization scheme, showing how deformation quantization and $SU(2)$ semiclassical limits describe off-planar fluctuations and connect to Richardson-Gaudin Hamiltonians. This approach provides a cohesive variational perspective on non-equilibrium synchronization and reveals deep links between Kuramoto dynamics, spin models, and integrable structures. The results open pathways to studying universality, integrability, and quantization in non-equilibrium phase transitions induced by synchronization.

Abstract

Despite being under intense scrutiny for 50 years, the Kuramoto oscillator model has remained a quintessential representative of non-equilibrium phase transitions. One of the reasons for its enduring relevance is the apparent lack of an optimization formulation, due to the fact that (superficially), the equations of motion seem to not be compatible with a Lagrangian structure. We show that, as a mean-field classical (twisted) spin model on $S^2$, the Kuramoto model can be described variationaly. Based on this result perturbation analysis around (unstable) Kuramoto equilibria are shown to be equivalent to low-energy fluctuations of mean-field Heisenberg spin models. Intriguingly, off-plane perturbations around these equilibria configurations turn out to be described by a semiclassical Gaudin model, pointing to the fact that oscillator synchronization maps to the spin pairing mechanism investigated by Richardson and subsequently by others.

The Lagrangian and symplectic structures of the Kuramoto oscillator model

TL;DR

This work establishes a variational framework for the Kuramoto oscillator by embedding it into a mean-field classical spin system on , deriving a corresponding Lagrangian and Hamiltonian that capture both planar and off-plane perturbations. Planar perturbations recover known synchronization stability results, while off-planar fluctuations are shown to be governed by a semiclassical Gaudin/Richardson spin-pairing structure, linking oscillator synchronization to spin-pairing physics. The authors further develop a geometric quantization scheme, showing how deformation quantization and semiclassical limits describe off-planar fluctuations and connect to Richardson-Gaudin Hamiltonians. This approach provides a cohesive variational perspective on non-equilibrium synchronization and reveals deep links between Kuramoto dynamics, spin models, and integrable structures. The results open pathways to studying universality, integrability, and quantization in non-equilibrium phase transitions induced by synchronization.

Abstract

Despite being under intense scrutiny for 50 years, the Kuramoto oscillator model has remained a quintessential representative of non-equilibrium phase transitions. One of the reasons for its enduring relevance is the apparent lack of an optimization formulation, due to the fact that (superficially), the equations of motion seem to not be compatible with a Lagrangian structure. We show that, as a mean-field classical (twisted) spin model on , the Kuramoto model can be described variationaly. Based on this result perturbation analysis around (unstable) Kuramoto equilibria are shown to be equivalent to low-energy fluctuations of mean-field Heisenberg spin models. Intriguingly, off-plane perturbations around these equilibria configurations turn out to be described by a semiclassical Gaudin model, pointing to the fact that oscillator synchronization maps to the spin pairing mechanism investigated by Richardson and subsequently by others.
Paper Structure (20 sections, 7 theorems, 100 equations)

This paper contains 20 sections, 7 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. A brief review of the Kuramoto model
  3. Oscillators interacting via a mean-field attractive force - a first analysis of the phase-locking and synchronization phenomena.
  4. Analyzing variational presentations of the Kuramoto model in two dimensions
  5. Non-Lagrangian nature of the original Kuramoto equations.
  6. A first Poisson bracket structure for Kuramoto oscillators.
  7. The special case of identical oscillators.
  8. A generalized Kuramoto model as a mean-field spin system in $\mathbb{R}^3$
  9. Global properties of the generalized Kuramoto model in $\mathbb{R}^3$
  10. The global synchronization condition and asymptotic states.
  11. Spin-flip dynamics on a stationary background.
  12. The complete synchronization limit.
  13. Lagrangian structure of the generalized Kuramoto model.
  14. A Hamiltonian structure for the generalized Kuramoto model
  15. Planar perturbation Hamiltonian around general synchronization solutions.
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\alpha \in \mathbb{R}$ and $u, v \in \mathbb{C}^*$. We list the following properties of the bilinear forms defined in Eq. eq_10:

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7