Three-Body Problem in Conformal-Euclidean Space: Complexity of a Low-Dimensional System

A general three-body problem is formulated on a curved geometry related to the energy surface of the system of bodies, which allows us to reveal hidden symmetries of the internal motion of a dynamical system and describe it by a system of stiff 6th-order ODEs instead of the usual 8th-order ones. In this formulation, the three-body problem is equivalent to the problem of propagation of a flow of geodesic trajectories on a 3D Riemannian manifold. A new criterion for the divergence of close geodesic trajectories is defined, similar to the Lyapunov exponent only on finite time intervals. Using the stochastic equation of motion of a system of bodies, a second-order partial differential equation of the Fokker-Planck type is derived for the probability distribution of geodesics (PDG) in phase space. Using PDG in a current tube, the entropy of a low-dimensional dynamical system is constructed and its complexity and disequilibrium are estimated. The behavior of new timing parameter (internal time) in global or 3D Jacobi space is studied in detail and its dimension is calculated.