Radiation Reaction and Center Manifolds

We study the effective dynamics of a mechanical particle coupled to a wave field and subject to the slowly varying potential V (εq) with ε small. To lowest order in ε the motion of the particle is governed by an effective Hamiltonian. In the next order one obtains “dissipative” terms which describe the radiation reaction. We establish that this dissipative dynamics has a center manifold which is repulsive in the normal direction and which is global, in the sense that for given data and sufficiently small ε the solution stays on the center manifold forever. We prove that the solution of the full system is well approximated by the effective dissipative dynamics on its center manifold.