Fundamental irreversibility of the classical three-body problem. New approaches and ideas in the study of dynamical systems

TL;DR

This paper demonstrates the fundamental irreversibility of the classical three-body problem by formulating it as geodesic flows on a Riemannian manifold, revealing hidden symmetries and deriving stochastic equations for external forces.

Contribution

It introduces a novel geometric approach to the three-body problem, reducing it to a 6th order system and establishing irreversibility through coordinate transformations and algebraic equations.

Findings

Geodesic equations reveal hidden symmetries.

Irreversibility is demonstrated via algebraic system reduction.

Stochastic equations describe external force influences.

Abstract

The three-body general problem is formulated as a problem of geodesic trajectories flows on the Riemannian manifold. It is proved that a curved space with local coordinate system allows to detect new hidden symmetries of the internal motion of a dynamical system and reduce the three-body problem to the system of 6\emph{th} order. It is shown that the equivalence of the initial Newtonian three-body problem and the developed representation provides coordinate transformations in combination with the underdetermined system of algebraic equations. The latter makes a system of geodesic equations relative to the evolution parameter, i.e., to the arc length of the geodesic curve, irreversible. Equations of deviation of geodesic trajectories characterizing the behavior of the dynamical system as a function of the initial parameters of the problem are obtained. To describe the motion of a dynamical system influenced by the external regular and stochastic forces, a system of stochastic equations (SDE) is obtained. Using the system of SDE, a partial differential equation of the second order for the joint probability distribution of the momentum and coordinate of dynamical system in the phase space is obtained. A criterion for estimating the degree of deviation of probabilistic current tubes of geodesic trajectories in the phase and configuration spaces is formulated. The mathematical expectation of the transition probability between two asymptotic subspaces is determined taking into account the multichannel character of the scattering.