On the Stability of Solitons for the Maxwell-Lorentz Equations with Rotating Particle

We prove the stability of solitons of the Maxwell–Lorentz equations with extended charged rotating particle. The solitons are solutions which correspond to the uniform rotation of the particle. To prove the stability, we construct the Hamilton–Poisson representation of the Maxwell–Lorentz system. The construction relies on the Hamilton least action principle. The constructed structure is degenerate and admits a functional family of the Casimir invariants. This structure allows us to construct the Lyapunov function corresponding to a soliton. The function is a combination of the Hamiltonian with a suitable Casimir invariant. The function is conserved, and the soliton is its critical point. The key point of the proof is a lower bound for the Lyapunov function. This bound implies that the soliton is a strict local minimizer of the function. The bound holds if the effective moment of inertia of the particle in the Maxwell field is sufficiently large with respect to the “bar moment of inertia".