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.
Highlights
One geometry cannot be more accurate than another, it may only be more convenient
We prove the equivalence of the original Newton three-body problem to the problem of geodesic flows on a Riemannian manifold
The main problem is to prove that the 6th order system (26) is equivalent to the original three-particle Newtonian problem (16). That both representations will be equivalent, if we prove that there exists continuous one-to-one mappings between the two following manifolds E6 and M, where E6 ⊂ R6 is a subspace allocated from the Euclidean space R6 taking into account the condition: g({ρ}) = E − V({ρ}) = 0
Summary
One of the important and insufficiently studied problems of the theory of collisions is the accurate account of the contribution of multichannel scattering to a specific elementary atomic-molecular process. As shown in a series of works [25,26,27,28], a representation developed on the basis of Riemannian geometry allows one to detect new hidden internal symmetries of dynamical systems The latter allows one to realize a more complete integration of the three-body problem, which in the general case in the sense of Poincaré is a non-integrable dynamical system. The proof of the irreversibility of the general three-body problem with respect to the internal time of the system allows us to solve the fundamentally important problem of quantum-classical correspondence for dynamical Poincaré systems. In Appendix part which includes Appendices A–G, provides important proof supporting the mathematical rigor of the developed approaches
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