In this paper,we study chaotic behavior of the(N+1)-body planar ring problem.Firstly,based on the perturbation theory of integrable Hamiltonian systems,we utilize mass ratio as a disturbance parameter so that this rin...In this paper,we study chaotic behavior of the(N+1)-body planar ring problem.Firstly,based on the perturbation theory of integrable Hamiltonian systems,we utilize mass ratio as a disturbance parameter so that this ring problem is regarded as a perturbation of the two-body problem.Then,by applying the extended Melnikov method,we address that there are transversal homoclinic orbits in the ring problem.Afterwards,since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle,we construct an invertible map f satisfying Conley–Moser condition and finally conclude that the ring problem possesses chaotic behavior of the Smale horseshoe type.展开更多
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. Lagran...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.展开更多
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...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.展开更多
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 t...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.展开更多
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.
For planar Newtonian 4-body problems with equal masses,we use variational methods to prove the existence of a non-collision periodic choreography solution such that all bodies move on a rose-type curve with three petals.
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 m...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.展开更多
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...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 <a href="#ref.1">[1]</a> 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.展开更多
Most studies of seismic noise cross-correlation （NCC） have focused on regional/continental scale imaging using empirical surface-wave Green＇s functions extracted from primary （0.05-0.08 Hz） and secondary （0.1-0....Most studies of seismic noise cross-correlation （NCC） have focused on regional/continental scale imaging using empirical surface-wave Green＇s functions extracted from primary （0.05-0.08 Hz） and secondary （0.1-0.16 Hz） microseisms. In this work, we present the NCC results at higher frequencies （〉0.5 Hz） from 6 months seismic noise recorded by a local array （ - 4 km aperture） deployed along the Calico fault in the Mojave Desert, California. Both fast and slow propagating waves are observed from the NCC record-sections. We compare the NCCs from sensor pairs that share a common sensor with the records of a borehole shot located very close to this common sensor. The result shows a good match of the slow surface-wave arrivals, indicating that the NCC method is able to recover unbiased surface-wave Green＇s functions at local scales. The strong body-wave NCC component is caused by the P waves generated offshore California. Along a SW-NE profile across the fault, we observe apparent P-wave arrivals and their reflections, which can be explained by a low-velocity- zone （LVZ） along the Calico fault. We calculat6 the LVZ width to be - 2.3 kin, and the P-wave velocity reduction within the LVZ to be -35 %. These estimates are consistent with other evidence for a relatively wide LVZ along the Calico fault.展开更多
Numerical simulation is an important tool that is helpful for us to understand the process of structure formation in the universe. However, many simulation results of cold dark matter （CDM） halos on a small scale ar...Numerical simulation is an important tool that is helpful for us to understand the process of structure formation in the universe. However, many simulation results of cold dark matter （CDM） halos on a small scale are inconsistent with observations： the central density profile is too cuspy and there are too many substructures, Here we point out that both the problems may be connected with a hitherto unrecognized bias in the simulated halos. Although CDM halos in nature and in simulation are both virialized systems of collisionless CDM particles, gravitational encounter cannot be neglected in the simulated halos because they contain many fewer particles. We demonstrate this by two numerical experiments, showing that there is a difference on the microcosmic scale between the natural and simulated halos. The simulated halo is more akin to globular clusters where gravitational encounter is known to lead to such drastic phenomena as core collapse. Such an artificial core collapse process appears to link the two problems together in the bottom-up scenario of structure formation in the ACDM universe. The discovery of this bias also has implications on the applicability of the Jeans theorem in galactic dynamics.展开更多
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.
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 ma...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.展开更多
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 p...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.展开更多
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 nu...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.展开更多
The nearest black hole to Earth,Sagittarius A^(*)(Sgr A^(*)),with its intense gravitational field,provides a unique opportunity to explore black hole mysteries.Over the past few decades,monitoring of the S2 star has p...The nearest black hole to Earth,Sagittarius A^(*)(Sgr A^(*)),with its intense gravitational field,provides a unique opportunity to explore black hole mysteries.Over the past few decades,monitoring of the S2 star has provided extensive valuable data that can be utilized to examine various gravity theories and black hole paradigms.This paper focuses on the most intriguing objects in astronomy,spinning black holes,and investigates the effects of spin on orbital motion.By applying the Markov Chain Monte Carlo algorithm to publicly available observational data of the S2 star,our findings indicate that current data fail to constrain the spin of Sgr A^(*).Simulated stars with smaller semi-major axes reveal that the direction of Lense-Thirring precession aligns with the spin direction of Sgr A^(*).Additionally,by incorporating the cosmological constant,which accounts for the expansion of the universe,into our analysis,we establish an upper limit of Λ■7.3×10^(-34) km^(-2) on the cosmological constant at the lo confidence level.Future long-term monitoring of S-cluster stars,combined with enhanced observational precision,may enable the determination of the spin of Sgr A^(*)and further tighten the bound on the cosmological constant.展开更多
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.
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.
基金supported by the National Natural Science Foundation of China(Grant No.11601257)Natural Science Foundation of Hebei Province(Grant Nos.A2019202342,A2023202041)。
文摘In this paper,we study chaotic behavior of the(N+1)-body planar ring problem.Firstly,based on the perturbation theory of integrable Hamiltonian systems,we utilize mass ratio as a disturbance parameter so that this ring problem is regarded as a perturbation of the two-body problem.Then,by applying the extended Melnikov method,we address that there are transversal homoclinic orbits in the ring problem.Afterwards,since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle,we construct an invertible map f satisfying Conley–Moser condition and finally conclude that the ring problem possesses chaotic behavior of the Smale horseshoe type.
基金Partially supported by the NNSF and MCME of China. the Qiu Shi Sci. and Tech. Foundation.Edn. Comm. of Tianjun CityAssociate Member of the ICTP.Partially supported by the NNSF of China
文摘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.
文摘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.
文摘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.
基金supported by National Natural Science Foundation of China(Grant No.11071175)a grant for advisor and PhD students from educational committee of China
文摘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.
基金supported by National Natural Science Foundation of China,a grant for Ph.D.of Chinese Educational Committeethe Scientific Research Foundation of Huaiyin Institute of Technology(Grant Nos.HGB0913,HGC0928)
文摘For planar Newtonian 4-body problems with equal masses,we use variational methods to prove the existence of a non-collision periodic choreography solution such that all bodies move on a rose-type curve with three petals.
基金The authors thank sincerely Professor Shanzhong Sun for his careful reading and helpful comments on the manuscript of this paper. The first author was partially supported by the Doctoral Innovation Project of Nankai University. The second author was partially supported by the National Natural Science Foundation of China （Grant No. 11131004）, MCME, LPMC of Ministry of Education of China, Nankai University, and the BCMIIS at Capital Normal University.
文摘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.
文摘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 <a href="#ref.1">[1]</a> 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.
文摘Most studies of seismic noise cross-correlation （NCC） have focused on regional/continental scale imaging using empirical surface-wave Green＇s functions extracted from primary （0.05-0.08 Hz） and secondary （0.1-0.16 Hz） microseisms. In this work, we present the NCC results at higher frequencies （〉0.5 Hz） from 6 months seismic noise recorded by a local array （ - 4 km aperture） deployed along the Calico fault in the Mojave Desert, California. Both fast and slow propagating waves are observed from the NCC record-sections. We compare the NCCs from sensor pairs that share a common sensor with the records of a borehole shot located very close to this common sensor. The result shows a good match of the slow surface-wave arrivals, indicating that the NCC method is able to recover unbiased surface-wave Green＇s functions at local scales. The strong body-wave NCC component is caused by the P waves generated offshore California. Along a SW-NE profile across the fault, we observe apparent P-wave arrivals and their reflections, which can be explained by a low-velocity- zone （LVZ） along the Calico fault. We calculat6 the LVZ width to be - 2.3 kin, and the P-wave velocity reduction within the LVZ to be -35 %. These estimates are consistent with other evidence for a relatively wide LVZ along the Calico fault.
文摘Numerical simulation is an important tool that is helpful for us to understand the process of structure formation in the universe. However, many simulation results of cold dark matter （CDM） halos on a small scale are inconsistent with observations： the central density profile is too cuspy and there are too many substructures, Here we point out that both the problems may be connected with a hitherto unrecognized bias in the simulated halos. Although CDM halos in nature and in simulation are both virialized systems of collisionless CDM particles, gravitational encounter cannot be neglected in the simulated halos because they contain many fewer particles. We demonstrate this by two numerical experiments, showing that there is a difference on the microcosmic scale between the natural and simulated halos. The simulated halo is more akin to globular clusters where gravitational encounter is known to lead to such drastic phenomena as core collapse. Such an artificial core collapse process appears to link the two problems together in the bottom-up scenario of structure formation in the ACDM universe. The discovery of this bias also has implications on the applicability of the Jeans theorem in galactic dynamics.
文摘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.
文摘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.
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China(12275034,12347101)。
文摘The nearest black hole to Earth,Sagittarius A^(*)(Sgr A^(*)),with its intense gravitational field,provides a unique opportunity to explore black hole mysteries.Over the past few decades,monitoring of the S2 star has provided extensive valuable data that can be utilized to examine various gravity theories and black hole paradigms.This paper focuses on the most intriguing objects in astronomy,spinning black holes,and investigates the effects of spin on orbital motion.By applying the Markov Chain Monte Carlo algorithm to publicly available observational data of the S2 star,our findings indicate that current data fail to constrain the spin of Sgr A^(*).Simulated stars with smaller semi-major axes reveal that the direction of Lense-Thirring precession aligns with the spin direction of Sgr A^(*).Additionally,by incorporating the cosmological constant,which accounts for the expansion of the universe,into our analysis,we establish an upper limit of Λ■7.3×10^(-34) km^(-2) on the cosmological constant at the lo confidence level.Future long-term monitoring of S-cluster stars,combined with enhanced observational precision,may enable the determination of the spin of Sgr A^(*)and further tighten the bound on the cosmological constant.
基金the National Natural Science Foundation of China
文摘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.
基金the National Natural Science Foundation of China.
文摘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.
