Horseshoes in a relativistic Hamiltonian system in 1+1 dimensions

We study the classical motion of a relativistic two-body system, in 1+1 dimensions, with interaction described by a relativistic generalization of the well-known Duffing potential. The equations of motion are separable in hyperbolic coordinates and are solved in quadrature. The radial equation (in the invariant variable corresponding to the spacelike distance between the particles) has an effective potential depending on the separation constant for the hyperbolic angular momentum, and analytic solutions are obtained for the separatrix motion. In the presence of weak driving and damping forces, the Melnikov criterion for the existence of homoclinic instability is applied, and it is shown that chaotic behavior is predicted for sufficiently strong driving forces (bounds are given).