An exactly soluble three-body problem in one dimension

1980 ◽  
Vol 58 (6) ◽  
pp. 719-728 ◽  
Author(s):  
C. Jung
Keyword(s):  
Closed Form ◽  
Bound States ◽  
Body Problem ◽  
Potential Functions ◽  
One Dimension ◽  
Three Body Problem ◽  
One Dimensional ◽  
Continuum States ◽  
Step Functions ◽  
Three Body

An exactly soluble one-dimensional three-body problem is presented, in which the interaction between the particles consists of local two-body potentials between each two particles. Infinitely high step functions are chosen for the form of the three potential functions. This interaction allows only three-body bound states and no continuum states. We have considered three different choices of the mass ratios of the three particles and we give formulas in closed form for the energies and for the wavefunctions of all states.

PERIODIC ORBITS NEAR A HETEROCLINIC LOOP FORMED BY ONE-DIMENSIONAL ORBIT AND A TWO-DIMENSIONAL MANIFOLD: APPLICATION TO THE CHARGED COLLINEAR THREE-BODY PROBLEM

2007 ◽  
Vol 17 (06) ◽  
pp. 2175-2183
Author(s):  
JAUME LLIBRE ◽  
DANIEL PAŞCA
Keyword(s):  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Negative Energy ◽  
Dimensional Manifold ◽  
Three Body Problem ◽  
Heteroclinic Loop ◽  
One Dimensional ◽  
Three Body ◽  
Total Collision

This paper is devoted to the study of a type of differential systems which appear usually in the study of the Hamiltonian systems with two degrees of freedom. We prove the existence of infinitely many periodic orbits on each negative energy level. All these periodic orbits pass near to the total collision. Finally we apply these results to study the existence of periodic orbits in the charged collinear three-body problem.

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

Particles ◽  
2020 ◽  
Vol 3 (3) ◽  
pp. 576-620
Author(s):  
A. S. Gevorkyan
Keyword(s):  
Dynamical System ◽  
Euclidean Space ◽  
Bound States ◽  
Body Problem ◽  
Local Coordinate System ◽  
Order System ◽  
Random Environments ◽  
Three Body Problem ◽  
Internal Time ◽  
Three Body

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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