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Self-interaction model of classical point particle in one-dimension

We consider a hamiltonian system on the real line, consisting of real scalar field $ϕ(x,t)$ and point particle with trajectory $y(t)$. The dynamics of this system is defined by the system of two equations: wave equation for the field, <<radiated>> by the point particle, and Newton&#39;s equation for the particle in its own field. We find the solution where the particle is strongly damped, but the kinetic and interaction energies of the field increase linearly in time, in despite of the full energy conservation.