  The article formulates the classical three-body problem in
conformal-Euclidean space (Riemannian manifold), and its equivalence to the
Newton three-body problem is mathematically rigorously proved. It is shown that
a curved space with a local coordinate system allows us to detect new hidden
symmetries of the internal motion of a dynamical system, which allows us to
reduce the three-body problem to the 6\emph{th} order system. A new approach
makes the system of geodesic equations with respect to the evolution parameter
of a dynamical system (\emph{internal time}) \emph{fundamentally irreversible}.
To describe the motion of three-body system in different random environments,
the corresponding stochastic differential equations (SDEs) are obtained. Using
these SDEs, Fokker-Planck-type equations are obtained that describe the joint
probability distributions of geodesic flows in phase and configuration spaces.
The paper also formulates the quantum three-body problem in conformal-Euclidean
space. In particular, the corresponding wave equations have been obtained for
studying the three-body bound states, as well as for investigating multichannel
quantum scattering in the framework of the concept of internal time. This
allows us to solve the extremely important quantum-classical correspondence
problem for dynamical Poincar\'e systems.
