微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量

把极角θ视为独立变量,得到Kepler系统的轨道微分方程.首先讨论Kepler系统轨道微分方程的Lie对称性和不变量,微扰Kepler系统轨道微分方程的精确Lie对称性和精确不变量,其次讨论微扰Kepler系统轨道微分方程的近似Lie对称性和近似不变量,并得到了微扰Kepler系统的9个一阶近似Lie对称性和6个一阶近似不变量,其中1个实为精确不变量,而其余5个分别等于微扰系数ε乘以Kepler系统相应的5个不变量。     

Binomial potential of the electron-proton interaction as an alternative to the Coulomb law

It is shown that a solution of the classical Kepler problem with a binomial potential explains the stationary orbits of the hydrogen atom, its discrete spectrum, and trajectories of scattered electrons with energies from several electron-volts to relativistic values. The Schrodinger equation with the binomial potential takes the form of a classical wave equation. Hence, the binomial potential is a more appropriate model for understanding atomic phenomena than the model based on the Coulomb potential.     

开普勒问题的矢量解法

直接利用矢量解法求解开普勒问题,不必借助于比奈方程．该方法简明扼要,物理图像清晰．     

Kepler方程的Noether对称性与Hojman守恒量

研究Kepler方程的Noether对称性与Hojman守恒量.给出系统的运动微分方程并给出Noether对称性的确定方程,提出Kepler方程的Noether对称性导致的Hojman守恒量.     

Unique distinction of the Lorentz and the anti-de Sitter group in the classical Kepler problem

In contrast to the familiar group-theoretical investigations of the Kepler problem, providing rather different symmetry structures for different types of state, the question is posed whether the problem does not exhibit uniquely distinguished dynamic invariance symmetries, which may be considered characteristic for the Newtonian interaction, independent of the special type of orbit. Taking into account only one Laplace vector it is shown that one, and only one, such symmetry exists and that this is just that of the Lorentz group. One is led to consider a distinguished orthonormal dynamical frame of the system, providing an extension of the Lorentz invariance symmetry to that of the anti-de Sitter group. Finally, some remarks are made concerning the relativistic formulation and solution of the Kepler problem.In memory of Reinhard Johannes Stickforth.     

Second-order systems with Runge-Lenz-type vectors

A class of second-order equation fields which contains the equations of motion of the Kepler and charge monopole problems is considered. In particular, the problems of the existence of a second integral of motion quadratic in momenta besides the energy is investigated and completely solved. A subclass of systems is exhibited which have a Runge-Lenz-type vector, but it is shown that the monopole system itself has no second (time-independent) quadratic integral of motion.     

Quantization of chaotic systems

Starting from the semiclassical dynamical zeta function for chaotic Hamiltonian systems we use a combination of the cycle expansion method and a functional equation to obtain highly excited semiclassical eigenvalues. The power of this method is demonstrated for the anisotropic Kepler problem, a strongly chaotic system with good symbolic dynamics. An application of the transfer matrix approach of Bogomolny is presented leading to a significant reduction of the classical input and to comparable accuracy for the calculated eigenvalues.     

Kepler motion on single-sheet hyperboloid

The classical Kepler–Coulomb problem on the single-sheeted hyperboloid H ³ ₁ is solved in the framework of the Hamilton–Jacobi equation. We have proven that all the bounded orbits are closed and periodic. The paths are ellipses or circles for finite motion.     

Structure of the generalized Friedmann problem

An investigation of those cases of the generalized Friedmann equation which are solvable in terms of elementary or elliptic functions is undertaken together with a study of the time gauges which allow this to occur. This is accomplished by examining the natural choices of independent and dependent variables in this problem using manipulations like those of the Kepler problem, which is shown to be equivalent to a generalized Friedmann problem, thus clarifying the similarities between the simplest solutions of each.     

A conservative discretization of the Kepler problem based on the L-transformations

Conservative numerical schemes for the two- and three-dimensional Kepler motion were recently proposed in [Y. Minesaki, Y. Nakamura, Phys. Lett. A 306 (2–3) (2002) 127; Y. Minesaki, Y. Nakamura, Phys. Lett. A 324 (4) (2004) 282]. They are based on Levi-Civita and Kustaanheimo–Stiefel transformations respectively. The schemes preserve all first integrals of Kepler motion: the Hamiltonian function, the angular momentum and Runge–Lenz vector. In the present Letter we extend this approach to the L-transformations (a generalization of Levi-Civita and Kustaanheimo–Stiefel transformations). Thus, we can consider more general family of the conservative numerical schemes for the two- and three-dimensional Kepler motion as well as construct conservative schemes for higher-dimensional Kepler problems. Conservation of the first integrals is proved with the help of L-matrix identities. The proposed numerical scheme permits explicit implementation.     

A strictly geometric interpretation of gravitation in general relativity

A geometric interpretation of gravitation is given using general relativity. The law of gravitation is taken in the formR ₄₄=0, whereR ₄₄is the component of the contracted Riemann-Christoffel (Ricci) tensor representing the curvature of time. The remaining curvature components of the contracted Riemann-Christoffel tensor may or may not vanish. All that is required in addition toR ₄₄=0 is that the Gaussian curvatureR be nowhere infinite. The conditionR ₄₄=0 yields a nonlinear wave equation. One of the static degenerate solutions represents the gravitational field surrounding a static gravitational point singularity. It is found that for this solution, the three famous predictions of general relativity are obtained in the weak-field approximation. In addition, it is found that there is a correction to the Kepler period of revolution for an orbit.     

Properties of Neutron Stars Rotating at Kepler Frequency with Uniform Strong Magnetic Field

A uniform strong magnetic field is considered in calculating the properties of neutron star rotating at the Kepler frequency. The results show that the effect of the magnetic field on the properties of neutron star is evident, and the properties of the neutron stars rotating at the Kepler frequency can be used as a criterion to the equations of states of the neutron star matters.     

The Kepler Problem with Allowance for Modification of the Gravitational Potential in the Expanding Universe with a Collisionless Gas

The paper deals with the solution to the Kepler problem for the gravitational potential of a point mass with allowance for the presence of collisionless particles in the Universe [3]. The estimates are reported for the Solar System planets and for the Sqr A object which is assumed to be a supermassive black hole at the center of the Milky Way [1]. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 26–28, August, 2005.     

Fractional extensions of the classical isotropic oscillator and the Kepler problem

The class of fractional Hamiltonian systems that generalize the classical problem of the two-dimensional (2D) isotropic harmonic oscillator and the Kepler problem is considered. It is shown that, in the 4D space of structural parameters, the 2D isotropic harmonic oscillator can be extended along a line in such a way that the orbits remain closed and oscillations remain isochronous. Likewise, the Kepler problem can be extended along a line in such a way that the orbits remain closed for all finite motions and the third Kepler law remains valid. These curves lie on the 2D surfaces where any dynamical system is characterized by the same rotation number for all finite motions.     

Stationary Kepler orbits of the electron with velocity-dependent perturbations — classical quantization

In this paper the evolution of Kepler orbits generated by velocity-dependent perturbations is discussed. It is found that in the presence of oscillatory perturbations, of oscillation frequency proportional to the kinetic energy of the moving particle, a discrete set of stationary orbits exists. If the coefficient of proportionality in the frequency—kinetic energy relation is the Planck constant, the orbits are the same as those given by Bohr's quantum postulates. The conclusion is drawn that the Schrödinger wave equation describes, in hidden form, velocity-dependent oscillatory perturbations superimposed upon the basic motion of an electron and has nothing in common with the basic trajectory of the motion.     

Periodic solutions of symmetric Kepler perturbations and applications

We investigate the existence of several families of symmetric periodic solutions as continuation of circular orbits of the Kepler problem for certain symmetric differentiable perturbations using an appropriate set of Poincaré-Delaunay coordinates which are essential in our approach. More precisely, we try separately two situations in an independent way, namely, when the unperturbed part corresponds to a Kepler problem in inertial cartesian coordinates and when it corresponds to a Kepler problem in rotating coordinates on ?³. Moreover, the characteristic multipliers of the symmetric periodic solutions are characterized. The planar case arises as a particular case. Finally, we apply these results to study the existence and stability of periodic orbits of the Matese-Whitman Hamiltonian and the generalized Størmer model.     

Geometric quantization of the five-dimensional Kepler problem

An extension of the Hurwitz transformation to a canonical transformation between phase spaces allows conversion of the five-dimensional Kepler problem into that of a constrained harmonic oscillator problem in eight dimensions. Thus a new regularization of the Kepler problem is established. Then, following Dirac, we quantize the extended phase space, imposing constraint conditions as superselection rules. In that way the interchangeability of the reduction and the quantization procedures is proved.     

Microlensing of Kepler stars as a method of detecting primordial black hole dark matter

If the dark matter consists of primordial black holes (PBHs), we show that gravitational lensing of stars being monitored by NASA's Kepler search for extrasolar planets can cause significant numbers of detectable microlensing events. A search through the roughly 150,000 light curves would result in large numbers of detectable events for PBHs in the mass range 5×10(-10) M(⊙) to 10(-4) M(⊙). Nondetection of these events would close almost 2 orders of magnitude of the mass window for PBH dark matter. The microlensing rate is higher than previously noticed due to a combination of the exceptional photometric precision of the Kepler mission and the increase in cross section due to the large angular sizes of the relatively nearby Kepler field stars. We also present a new formalism for calculating optical depth and microlensing rates in the presence of large finite-source effects.     

Generalisations of the Laplace–Runge–Lenz Vector

Abstract

The characteristic feature of the Kepler Problem is the existence of the so-called Laplace–Runge–Lenz vector which enables a very simple discussion of the properties of the orbit for the problem. It is found that there are many classes of problems, some closely related to the Kepler Problem and others somewhat remote, which share the possession of a conserved vector which plays a significant rôle in the analysis of these problems.