Reduction in the two-body problem on the Lobatchevsky plane

We present a reduction-of-order procedure in the problem of motion of two bodies on the Lobatchevsky plane $\mathbb H^2$. The bodies interact with a potential that depends only on the distance between the bodies (this holds for an analog of the Newtonian potential). In earlier works, this reduction procedure was used to analyze the motion of two bodies on the sphere.