Abstract:

The article formulates the classical three-body problem in conformal-Euclidean space (Riemannian manifold), and its equivalence to the Newton three-body problem is mathematically rigorously proved. It is shown that a curved space with a local coordinate system allows us to detect new hidden symmetries of the internal motion of a dynamical system, which allows us to reduce the three-body problem to the 6th order system. A new approach makes the system of geodesic equations with respect to the evolution parameter of a dynamical system (internal time) fundamentally irreversible. To describe the motion of three-body system in different random environments, the corresponding stochastic differential equations (SDEs) are obtained. Using these SDEs, Fokker-Planck-type equations are obtained that describe the joint probability distributions of geodesic flows in phase and configuration spaces. The paper also formulates the quantum three-body problem in conformal-Euclidean space. In particular, the corresponding wave equations have been obtained for studying the three-body bound states, as well as for investigating multichannel quantum scattering in the framework of the concept of internal time. This allows us to solve the extremely important quantum-classical correspondence problem for dynamical Poincaré systems

Date accepted:

27 July 2020

Date of publication:

4 August 2020

Identifier:

oai:noad.sci.am:136245

DOI:

10.3390/particles3030039

Language:

English

Journal or Publication Title:

Particles

Volume:

3

Number:

3

URL:

Affiliation:

Institute for Informatics and Automation Problems of NAS RA ; Institute of Chemical Physics, NAS of Armenia,

Year:

2020