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.     

Symmetries and Rank Deficiency in the Orbit Determination Around Another Planet

We study the possible degeneracies for the normal matrix of the observations from the Earth of the motion of a satellite around a planet, and give the possible solutions to the loss of precision in the orbit determination caused by the rank deficiency. Finally we discuss the methods available to control the instability in the orbit determination resulting from the degeneracy.     

Observations on seismic wave equation and reflection coefficient symmetries in stratified media

Summary. The equations describing seismic waves in a stratified earth have a number of symmetry properties, one of which has recently been used by Garmany to derive a simple expression for the inverse of the matrix of eigenvectors appearing in the solution of the equations. We review these symmetries of the wave equation in several notations to demonstrate that the property used by Garmany is distinct from the usual symmetries found in the seismological literature. Like the others, the new symmetry has implications for the reflection and transmission properties of a medium. These implications are briefly reviewed in order to show how the new symmetry is placed relative to the others. A limited discussion of the physical origins of the symmetries is given and, though the overall picture is incomplete, it is noted that the new symmetry yields conservation of energy for reflection/transmission at a single interface in all types of media (anisotropic, lossy, etc.).     

Symmetries of time-dependent magnetoconvection

Abstract

In the presence of a magnetic field, convection may set in at a stationary or an oscillatory bifurcation, giving rise to branches of steady, standing wave and travelling wave solutions. Numerical experiments provide examples of nonlinear solutions with a variety of different spatiotemporal symmetries, which can be classified by establishing an appropriate group structure. For the idealized problem of two-dimensional convection in a stratified layer the system has left-right spatial symmetry and a continuous symmetry with respect to translations in time. For solutions of period P the latter can be reduced to Z ₂ symmetry by sampling solutions at intervals of ½P. Then the fundamental steady solution has the spatiotemporal symmetry D ₂ = Z ₂ ? Z ₂ and symmetry-breaking yields solutions with Z ₂ symmetry corresponding to travelling waves, standing waves and pulsating waves. A further loss of symmetry leads to modulated waves. Interactions between the fundamental and its first harmonic are described by the group D _2h = D ₂ ? Z ₂ and its invariant subgroups, which describe solutions that are either steady or periodic in a uniformly moving frame. For a Boussinesq fluid in a layer with identical top and bottom boundary conditions there is also an up-down symmetry. With fixed lateral boundaries the spatiotemporal symmetries, again described by D _2h and its invariant subgroups, can be related to results obtained in numerical experiments and analysed by Nagata et al. (1990). With periodic boundary conditions, the full symmetry group, D _2h ?Z ₂, is of order 16. Its invariant subgroups describe pure and mixed-mode solutions, which may be steady states, standing waves, travelling waves, pulsating waves or modulated waves.     

Exact solutions of the Lane–Emden-type equation

We classify the Lane–Emden-type equation xy^″+ny^′+x^νf(y)=0 with respect to the standard Lagrangian according to the Noether point symmetries it admits. First integrals of the various cases, which admit Noether point symmetries, and reduction to quadratures for these cases are obtained. Six cases result in new solutions.     

A TEM and RUM study of the inherent displacive flexibility of the fresnoite framework structure type

 A careful electron diffraction study has been made of the incommensurately modulated room-temperature phases of the fresnoites Ba₂TiGe₂O₈ (BTG) and Ba₂TiSi₂O₈ (BTS) and used to determine their (3+1)- and (3+2)-dimensional superspace group symmetries. The primitive primary modulation wave vectors in both materials are found to occur close to the same position in the parent Brillouin zone, near ∼0.3〈110〉 _p ^*+1/2c _p ^st . A rigid unit mode (RUM) analysis of the inherent displacive structural flexibility of the ideal fresnoite framework structure type is then carried out in an attempt to understand the significance of this particular modulation wave vector. Six zero-frequency RUM modes and two close to zero frequency quasi-RUM (Q-RUM) modes are found to exist for any modulation wave vector. These RUM modes are all primarily associated with rotations of the constituent TO₄ (T=Si or Ge) tetrahedra and TiO₅ square pyramids around in-plane i.e. perpendicular to c rotation axes. A seventh RUM mode involving rotation of the constituent rigid polyhedra around c combined with shifts in the basal plane is found but only at a very specific modulation wave vector q∼0.30〈110〉 _p ^* , in close agreement with the condensed RUM mode found in the electron diffraction study. It is the condensation of just such a RUM mode that appears to play a major role in the various incommensurately modulated structures observed in Ba₂TiGe₂O₈, Ba₂TiSi₂O₈ and Sr₂TiSi₂O₈, respectively. Received: 26 November 2001 / Accepted: 25 May 2002 Present address: Y. Tabira Materials Characterization Laboratory Mitsui Mining & Smelting Co. Ltd. 1333-2 Haraichi, Ageo, Saitama 362-0021, Japan     

On M-Theory

This contribution gives a personal view on recent attempts to find a unified framework for non-perturbative string theories, with special emphasis on the hidden symmetries of supergravity and their possible role in this endeavor. A reformulation ofd = 11 supergravity with enlarged tangent space symmetry SO(1, 2) × SO(16) is discussed from this perspective, as well as an ansatz to construct yet further versions with SO(1, 1) × SO(16)^∞ and possibly even SO(1, 1)₊ × ISO(16)^∞ tangent space symmetry. It is suggested that upon “third quantization”, dimensionally reduced maximal supergravity may have an equally important role to play in this unification as the dimensionally reduced maximally supersymmetricSU(∞) Yang Mills theory.     

Potential Symmetry Properties of a Family of Equations Occuring in Ice Sheet Dynamics

In this paper we derive some similarity solutions of a nonlinear equation associated with a free boundary problem arising in the shallow-water approximation in glaciology. In addition we present a classical potential symmetry analysis of this second-order nonlinear degenerate parabolic equation related to non-Newtonian ice sheet dynamics in the isothermal case. After obtaining a general result connecting the thickness function of the ice sheet and the solution of the nonlinear equation (without any unilateral formulation), a particular example of a similarity solution to a problem formulated with Cauchy boundary conditions is described. This allows us to obtain several qualitative properties on the free moving boundary in the presence of an accumulation-ablation function with realistic physical properties.     

Symmetric Periodic Orbits in the Anisotropic Schwarzschild-Type Problem

Studying the two-body problem associated to an anisotropic Schwarzschild-type field, Mioc et al. (2003) did not succeed in proving the existence or non-existence of periodic orbits. Here we answer this question in the affirmative. To do this, we start from two basic facts: (1) the potential generates a strong force in Gordon’s sense; (2) the vector field of the problem exhibits the symmetries S _i , , which form, along with the identity, an Abelian group of order 8 with three generators of order 2. Resorting to S ₂ and S ₃, in connection with variational methods (particularly the classical lower-semicontinuity method), we prove the existence of infinitely many S ₂- or S ₃-symmetric periodic solutions. The symmetries S ₂ and S ₃ constitute an indicator of the robustness of the classical isotropic Schwarzschild-type system to perturbations (as the anisotropy may be considered).