On reduction of the general three-body Newtonian problem and the curved geometry

Gevorkyan, A. A.

doi:10.1088/1742-6596/496/1/012030

Cited by 5 publications

(21 citation statements)

References 7 publications

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“…Lastly important to note that the full quantum theory of multichannel scattering in the three-body system allowing the emergence of quantum chaos in the wave function, can be constructed based on Schrödinger equation with the Hamiltonian (16), also taking into account the classical equations (15), (24) and (26). Recall that the system of the classical equations in this case is responsible for the topological peculiarities of tubes of the quantum probabilistic currents and transitions between asymptotic channels.…”

Section: Discussionmentioning

confidence: 99%

“…Let us assume that the dynamical system at the movement undergoes to the influence random forces, in particular to quantum fluctuations. In a mathematical sense, it is equivalent to the fact that the metric tensor and the corresponding coefficients in equations (15) are random functions: influences, the set of functions {η 0 (s), ..., η 3 (s)} denote random generators which will be refined below. It is obvious that the random component ofã i by a value is much more than a random member in theΛ, sinceã i is the first derivative of metric tensor.…”

Section: Classical Movement Under the Influence Of Quantum Fluc-tumentioning

confidence: 99%

“…With consideration of said the system of equations (15) can be expanded and presented in the form of stochastic equations of Langevin type:…”

Section: Classical Movement Under the Influence Of Quantum Fluc-tumentioning

confidence: 99%

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On the motion of classical three-body system with consideration of quantum fluctuations

Gevorkyan

2017

Phys. Atom. Nuclei

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We obtained the system of stochastic differential equations which describes the classical motion of the three-body system under influence of quantum fluctuations. Using SDEs, for the joint probability distribution of the total momentum of bodies system were obtained the partial differential equation of the second order. It is shown, that the equation for the probability distribution is solved jointly by classical equations, which in turn are responsible for the topological peculiarities of tubes of quantum currents, transitions between asymptotic channels and, respectively for arising of quantum chaos.

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Section: Discussionmentioning

confidence: 99%

Section: Classical Movement Under the Influence Of Quantum Fluc-tumentioning

confidence: 99%

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On the motion of classical three-body system with consideration of quantum fluctuations

Gevorkyan

2017

Phys. Atom. Nuclei

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“…In this work we significantly develop the above geometric and other ideas for studying the classical and quantum three-body problem in order to find new theoretical and algorithmic possibilities for the effective solution of these problems. Unlike previous authors, we solved the complex problem of mapping Euclidean geometry to Riemann geometry, which allowed us to make the theory consistent and mathematically rigorous [25]. In other words, we prove the equivalence of the original Newton three-body problem to the problem of geodesic flows on a Riemannian manifold.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

New Concept for Studying the Classical and Quantum Three-Body Problem: Fundamental Irreversibility and Time’s Arrow of Dynamical Systems

Gevorkyan

2020

Particles

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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.

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

Gevorkyan¹

2017

Preprint

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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 6th 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 differential 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.

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On reduction of the general three-body Newtonian problem and the curved geometry

Cited by 5 publications

References 7 publications

On the motion of classical three-body system with consideration of quantum fluctuations

On the motion of classical three-body system with consideration of quantum fluctuations

New Concept for Studying the Classical and Quantum Three-Body Problem: Fundamental Irreversibility and Time’s Arrow of Dynamical Systems

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

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