On stability of solitons for the Maxwell–Lorentz system with spinning particle

We consider stability of solitons of the Maxwell–Lorentz system with extended charged spinning particle. The solitons are solutions which correspond to a particle moving with a constant velocity v∈R3 with |v| < 1 and rotating with a constant angular velocity ω ∈ R3. Our main results are the orbital stability of moving solitons with ω = 0 and a linear orbital stability of rotating solitons with v = 0. The Hamilton–Poisson structure of the Maxwell–Lorentz system is degenerate and admits the Casimir invariants. We construct the Lyapunov function as a linear combination of the Hamiltonian with a suitable Casimir invariant. The key point is a lower bound for this function. The proof of the bound in the case ω ≠ 0 relies on angular momentum conservation and suitable spectral arguments including the Heinz–Löwner–Kato inequality and closed graph theorem.