A completely integrable case in three-particle problems with homogeneous potentials (Q1107189)

The equations of motion for classical three-body problems in one- dimensional case are treated with the potential which is a function homogeneous with respect to the coordinates. The author shows that if the total energy is zero, the equations can be reduced to first-order differential equation, there is an additional first integral, and the Hamilton-Jacobi equations are separable, and, therefore, that the equations of motion are integrable for this case. The author also shows that the invariant manifold of solutions is diffeomorphic to a two- dimensional torus.