Symplectic reduction, geometric phase, and internal dynamics in three-body molecular dynamics

Lin, Fei

doi:10.1016/s0375-9601(97)00608-7

Cited by 4 publications

(3 citation statements)

References 34 publications

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“…The general three-body classical problem is one of the oldest and most complex problems in classical mechanics [1][2][3][4][5][6]. Briefly, the meaning of the task is to study the motion of three bodies in space under the influence of pairwise interactions of bodies in accordance with Newton's theory of gravitation.…”

Section: A Poincarémentioning

confidence: 99%

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

Gevorkyan¹

2020

Preprint

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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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Section: A Poincarémentioning

confidence: 99%

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

Gevorkyan¹

2020

Preprint

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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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“…The three-body problem is one of the oldest and most complex problem in classical mechanics [2][3][4][5][6]. However, despite this, the study of this problem is still relevant in connection with its wide application in solving various applied problems, from celestial mechanics to the atom-molecular collision.…”

Section: Introductionmentioning

confidence: 99%

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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Symplectic reduction, geometric phase, and internal dynamics in three-body molecular dynamics

Cited by 4 publications

References 34 publications

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

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

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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