Abstract:
We show how to construct the hyperbolic plane with its geodesic flow
as the reduction of a three-problem whose potential
is proportional to $I/\Delta^2$ where $I$ is the moment of inertia of this
triangle whose vertices are the locations of the three bodies and $\Delta$ is its area.
The reduction method follows [11].
Reduction by scaling is only possible because the
potential is homogeneous of degree $-2$. In trying to extend the assertion
of hyperbolicity to the analogous family of
planar N-body problems with three-body interaction potentials we run into Mnëv's astounding universality theorem
which implies that the extended assertion is doomed to fail.
References