Particles, Vol. 7, Pages 1038-1061: General Three-Body Problem in Conformal-Euclidean Space: New Properties of a Low-Dimensional Dynamical System

Particles doi: 10.3390/particles7040063

Authors:
Ashot S. Gevorkyan
Aleksander V. Bogdanov
Vladimir V. Mareev

Despite the huge number of studies of the three-body problem in physics and mathematics, the study of this problem remains relevant due to both its wide practical application and taking into account its fundamental importance for the theory of dynamical systems. In addition, one often has to answer the cognitive question: is irreversibility fundamental for 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 has been proven 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 sixth-order system instead of the eighth order. An important consequence of the developed representation is that the chronologizing parameter of the motion of a system of bodies, which we call internal time, differs significantly from ordinary time in its properties. In particular, it more accurately describes the irreversible nature of multichannel scattering in a three-body system and other chaotic properties of a dynamical system. The paper derives an equation describing 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 dynamical system and the dimension of stochastic fractal structures arising in three-dimensional space are obtained. An effective mathematical algorithm is developed for the numerical simulation of the general three-body problem, which is traditionally a difficult-to-solve system of stiff ordinary differential equations.

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Ashot S. Gevorkyan www.mdpi.com