Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R^3 or R^2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular rostricted 3-body problem in R^3 or R^2 are also planar equilateral triangle circular solutions.

Geometric characterizations for variational minimization solutions of the 3-body problem

For any given positive masses it is proved that the variational minimization solutions of the 3-body problem in 3 or 2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in 3 or 2 are also planar equilateral triangle circular solutions.

Global Geometry of 3-Body Motions with Vanishing Angular MomentumⅠ

Following Jacobi＇s geometrization of Lagrange＇s least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is the conformal modification of the kinematic metric by the factor （U ＋ h）, where U and h are the potential function and the total energy, respectively. In the special case of 3-body motions with zero angular momentum, the global geometry of such trajectories can be reduced to that of their moduli curves, which record the change of size and shape, in the moduli space of oriented m-triangles, whose kinematic metric is, in fact, a Riemannian cone over the shape space M^＊≌S^2 （1/2）. In this paper, it is shown that the moduli curve of such a motion is uniquely determined by its shape curve （which only records the change of shape） in the case of h≠0, while in the special case of h = 0 it is uniquely determined up to scaling. Thus, the study of the global geometry of such motions can be further reduced to that of the shape curves, which are time-parametrized curves on the 2-sphere characterized by a third order ODE. Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonieity, that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems.

Variational minimizing parabolic and hyperbolic orbits for the restricted 3-body problems Dedicated to my Teacher Professor Yang Wannian on the Occasion of his 75th Birthday

Using variational minimizing methods,we prove the existence of the odd symmetric parabolic or hyperbolic orbit for the restricted 3-body problems with weak forces.

Geometric characterizations for variational minimizing solutions of charged 3-body problems

We study the charged 3-body problem with the potential function being （-a）-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the minimizers. We prove that if the charged 3-body problem admits a triangular central configuration, then the variational minimizing solutions of the problem in the τ/2-antiperiodic function space are exactly defined by the circular motions of this triangular central configuration.

Dominant Correlation Effects in Two-Electron Atoms

Two-electron atoms have been investigated near threshold of double escape within the framework of hyperspherical coordinates. A particularly useful set of hyperspherical angles has been used. It is well known for many years that the hyperradial motion is nearly separable from the hyperspherical angular motion. Therefore, the Born-Oppenheimer separation method should be useful. However, the success of that method in molecular physics is based on the small mass ratio, electron mass to nuclear mass. In the atomic application such a small parameter does not exist. Nevertheless the method works surprisingly well in the lower part of the spectrum. For increasing excitation energy the method becomes shaky. Near ionization threshold, it breaks even down. The author will present elsewhere an improved Born-Oppenheimer method. First pilot developments and comparison with the experimental situation are presented already here. Inclusion of a momentum-momentum radial coupling delivers an improved basis. We show that our extended Born-Oppenheimer approach leads to a deformation of the whole potential energy surface during the collision. In consequence of this deformation we outline a quantum derivation of the Wannier threshold cross section law, and we show that (e, 2e) angular distribution data are strongly influenced by that surface deformation. Finally, we present a mechanism for electron pair formation and decay leading to a supercurrent independent of the temperature. Our framework can be extended to more than two electrons, say 3 or 4. We conclude that our improved Born-Oppenheimer method [1] is expected not only to deliver better numerical data, but it is expected to describe also the Wannier phenomenon. The idea of the new theory together with first qualitative results is presented in this paper.

Molecular Integral for the Double Electron-Repulsion Potential： A Closed Form Underived for Forty Years

A closed form of the title integral formula over the Gaussian-type orbitals is derived for the first time. The present closed form is analytical as the multiple hyper-geometric function of five variables.

A Structural Study of High Cr Cast Iron as a Grinding Ball Material

In present paper, the 3-body abrasion and impact fatigue resistance of a 12％Cr-2.65％C-1.4％Si high Cr cast iron are comprehensively evaluated. The results indicated that the lower the C content of the martensitic matrix, the better the impact fatigue resistance of the iron. The retained austenite is always harmful to both 3-body abrasion and impact fatigue resistances. The low C content martensitic matrix free from retained austenite is suitable for making grinding balls.

Long-Range Correlation in Positron-Hydrogen Scattering System near the Threshold of <i>Ps</i>(<i>n</i>= 2) Formation

This is a short report of a recently uncovered resonant phenomenon. The modified Faddeev equation that correctly includes all six open channels is used. The calculation is carried out in s-partial wave. We report a number of resonant peaks in the elastic cross sections as well as the wave amplitudes involved. This is the energy region where the Stark-effect induced electric dipole energy split in the target dominates the physics and the Long-Range behavior of the 3-body scattering system. It is found that when the center of mass collision energy in the new channels is in integer proportion to the corresponding electric dipole energy split, Bremsstrahlung photon mediated resonant scattering occurs. The corresponding wave amplitudes deform into wave-packets hundreds to thousands of Bohr radii in width. The physical implication of this phenomenon will be discussed.

A Possible Role of the Gailitis Resonance in Muon Catalyzed Fusion

In a previous JMP article published May 2013, a comprehensive calculation was presented for all properties of a number of long-life s-state Gailitis resonances lying just above the PS(n = 2) formation threshold in a positron-Hydrogen scattering system. The six open-channel calculation was carried out by solving a set of four hundred thousand coupled linear equations. The modified Faddeev equation was used to obtain the wave-amplitude for each of the six open channels. Details can be found in reference [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400300037003200380034003600370034000000 . This note presents some qualitative properties of Gailitis resonances in the scattering systems of d + tu just above the thresholds of the opening of a new channel of the muonic atoms tu(n) or du(n), n > 2 is the principal quantum number. u is a negatively charged muon, d and t are the nuclei of the two isotopes of the Hydrogen atom with one and two neutrons in the nucleus respectively. We study the possible decay channels of some of the long-life Gailitis resonances. Of particular interest is a transition directly from a Gailitis (3-body) resonance to the bound states dtu molecular ions via a radiative emission of a photon or an external auger ejection of a nearby electron. Possible experimental evidence will be presented.

An analysis of the symmetry in 3-boson systems

The constraints arising from the quantum mechanical symmetry on wavefunctions, and the effect of the constraints on the structures and internal notions of quantum states are studied.

Effect of symmetry on structures and internal motions of 3-fermion systems

The effect of the quantum mechanical symmetry on the structures and internal motions is investigated. The results obtained are compared with those of boson systems.

The scaling behavior of the second virial coefficient of linear and ring polymer

The scaling behavior of the second virial coefficient of ring polymers at the theta temperature of the corresponding linear polymer(θ_L) is investigated by off-lattice Monte Carlo simulations. The effects of the solvents are modeled by pairwise interaction between polymer monomers in this approach. Using the umbrella sampling, we calculate the effective potential U(r) between two ring polymers as well as the second virial coefficient A_2 of ring polymers at θ_L, which results from a combination of 3-body interactions and topological constraints. The trend in the strength of the effective potential with respect to chain length shows a non-monotonic behavior, differently from that caused only by topological constraints. Our simulation suggests that there are three regimes about the scaling behavior of A_2 of ring polymers at θ_L: 3-body interactions dominating regime, the crossover regime, and the topological constraints dominating regime.