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 new 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 the new formulation, the three-body problem is reduced to the problem of propagating a flow of geodesic trajectories on a 3D Riemannian manifold. A new criterion for the divergence of close geodesic trajectories, similar to the Lyapunov exponent, is defined, and a second-order PDE 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 (deviation from the equilibrium probability distribution) are estimated.