Example of relativistic two-body problem. I. Boundary-value problem for minimal surface

A certain type of interaction of two particles is reduced to the Lagrange system of partial differential equations for the world surface connecting the world lines which are the particle trajectories, the mechanics equations being the boundary conditions. The simplest example of this type, the nonrelativistic model, is analysed. The two-body relativistic problem analogous to the considered nonrelativistic model also belongs to the type under consideration.