The General Three-Body Problem in Conformal-Euclidean Space: Hidden Symmetries and New Properties of a Low-Dimensional Syste

TL;DR

This paper reformulates the three-body problem in conformal Euclidean space, revealing hidden symmetries, reducing its complexity, and demonstrating the irreversibility of internal time, with implications for understanding dynamical systems and entropy.

Contribution

It introduces a novel formulation of the three-body problem in conformal space, uncovering hidden symmetries and establishing the irreversibility of internal time in classical dynamics.

Findings

Reduction of the problem from 8th to 6th order system

Discovery of hidden symmetries in the configuration space

Development of an algorithm for simulating the three-body problem

Abstract

Despite the huge number of research into the three-body problem in physics and mathematics, the study of this problem still remains relevant both from the point of view of its broad application and taking into account its fundamental significance for the theory of dynamical systems. In addition, to solve the problem of quantum-to-classical transition, it is important to answer the question: is irreversibility fundamental to the description of the classical world? To answer this question, we considered a reference classical dynamical system, the general three-body problem, formulating it in conformal Euclidean space and rigorously proving its equivalence to the Newtonian three-body problem. It is shown that a curved configuration space with a local coordinate system reveals new hidden symmetries of the internal motion of a dynamical system, which makes it possible to reduce the problem to a 6th order system instead of the known 8th order. The most important consequence of this consideration is that the chronologizing parameter of the motion of a system of particles, which we call internal time, is in the general case irreversible, which is characteristic of the general three-body problem. An equation is derived that describes the evolution of the flow of geodesic trajectories, with the help of which the entropy of the system is constructed. New criteria for assessing the complexity of a low-dimensional dynamic system and the dimension of stochastic fractal structures arising in three-dimensional space are obtained. An effective mathematical algorithm has been developed for the numerical simulation of the general three-body problem, which is traditionally a difficult-to-solve system of stiff ordinary differential equations.