On the Flag Curvature of Invariant Randers Metrics

In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field.      

Invariant solutions to the conformal Killing–Yano equation on Lie groups

We search for invariant solutions of the conformal Killing–Yano equation on Lie groups equipped with left invariant Riemannian metrics, focusing on 2-forms. We show that when the Lie group is compact equipped with a bi-invariant metric or 2-step nilpotent, the only invariant solutions occur on the 3-dimensional sphere or on a Heisenberg group. We classify the 3-dimensional Lie groups with left invariant metrics carrying invariant conformal Killing–Yano 2-forms.     

A spectral invariant representation of spectral reflectance

Spectral image acquisition as well as color image is affected by several illumination factors such as shading, gloss, and specular highlight. Spectral invariant representations for these factors were proposed for the standard dichromatic reflection model of inhomogeneous dielectric materials. However, these representations are inadequate for other characteristic materials like metal. This paper proposes a more general spectral invariant representation for obtaining reliable spectral reflectance images. Our invariant representation is derived from the standard dichromatic reflection model for dielectric materials and the extended dichromatic reflection model for metals. We proof that the invariant formulas for spectral images of natural objects preserve spectral information and are invariant to highlights, shading, surface geometry, and illumination intensity. It is proved that the conventional spectral invariant technique can be applied to metals in addition to dielectric objects. Experimental results show that the proposed spectral invariant representation is effective for image segmentation.     

An exact magnetic-moment invariant of charged-particle gyromotion

For the motion of a charged particle in a uniform, time-dependent axial magnetic field B(t)e(z), it is shown that there is an exact magnetic-moment invariant of the particle dynamics M, to which the adiabatic magnetic-moment invariant mu = mv2 perpendicular/2B is asymptotic when the time scale of the magnetic field variation is much slower than the gyroperiod. The connection between the exact invariant M and the adiabatic invariant mu enables us to characterize in detail the robustness of the adiabatic magnetic-moment invariant mu.     

Dynamical Invariant and Exact Mechanical Analyses for the Caldirola–Kanai Model of Dissipative Three Coupled Oscillators

We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators.     

Operator Derivation of the Gauge-Invariant Proca and Lehnert Equations; Elimination of the Lorenz Condition

Using covariant derivatives and the operator definitions of quantum mechanics, gauge invariant Proca and Lehnert equations are derived and the Lorenz condition is eliminated in U(1) invariant electrodynamics. It is shown that the structure of the gauge invariant Lehnert equation is the same in an O(3) invariant theory of electrodynamics.     

Group Invariant Solutions Without Transversality

We present a generalization of Lie's method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations. The characterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics, harmonic maps, and general relativity. This work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions. Received: 16 September 1999 / Accepted: 4 February 2000     

Stalactite basin structure of dynamical systems with transient chaos in an invariant manifold

Dynamical systems with invariant manifolds occur in a variety of situations (e.g., identical coupled oscillators, and systems with a symmetry). We consider the case where there is both a nonchaotic attractor (e.g., a periodic orbit) and a nonattracting chaotic set (or chaotic repeller) in the invariant manifold. We consider the character of the basins for the attracting nonchaotic set in the invariant manifold and another attractor not in the invariant manifold. It is found that the boundary separating these basins has an interesting structure: The basin of the attractor not in the invariant manifold is characterized by thin cusp shaped regions ("stalactites") extending down to touch the nonattracting chaotic set in the invariant manifold. We also develop theoretical scalings applicable to these systems, and compare with numerical experiments. (c) 2000 American Institute of Physics.     

二维各向同性变频率谐振子的超对称性及其不变量的超对称量子力学精确解

采用超对称量子力学与不变量相结合的方法讨论了二维各向同性变频率谐振子,给出了二维各向同性变频率谐振子的不变量,采用超对称量子力学方法精确求解了不变量的本征值和本征函数,并且给出了当频率恒定时,二维常频率谐振子的本征值和本征函数的精确解.最后对不变量的超对称性进行了讨论.     

Topological Quantum Field Theory for the Universal Quantum Invariant

We extend the universal quantum invariant defined in [15] to an invariant of 3-manifolds with boundaries, and show that the invariant satisfies modified axioms of TQFT. Received: 22 August 1996 / Accepted: 17 January 1997     

Renormalization-group analysis for the transition to chaos in Hamiltonian systems

We study the stability of Hamiltonian systems in classical mechanics with two degrees of freedom by renormalization-group methods. One of the key mechanisms of the transition to chaos is the break-up of invariant tori, which plays an essential role in the large scale and long-term behavior. The aim is to determine the threshold of break-up of invariant tori and its mechanism. The idea is to construct a renormalization transformation as a canonical change of coordinates, which deals with the dominant resonances leading to qualitative changes in the dynamics. Numerical results show that this transformation is an efficient tool for the determination of the threshold of the break-up of invariant tori for Hamiltonian systems with two degrees of freedom. The analysis of this transformation indicates that the break-up of invariant tori is a universal mechanism. The properties of invariant tori are described by the renormalization flow. A trivial attractive set of the renormalization transformation characterizes the Hamiltonians that have a smooth invariant torus. The set of Hamiltonians that have a non-smooth invariant torus is a fractal surface. This critical surface is the stable manifold of a single strange set encompassing all irrational frequencies. This hyperbolic strange set characterizes the Hamiltonians that have an invariant torus at the threshold of the break-up. From the critical strange set, one can deduce the critical properties of the tori (self-similarity, universality classes).     

Radial drift invariant in long-thin mirrors

In omnigenous systems, guiding centers are constrained to move on magnetic surfaces. Since a magnetic surface is determined by a constant radial Clebsch coordinate, omnigeneity implies that the guiding center radial coordinate (the Clebsch coordinate) is a constant of motion. Near omnigeneity is probably a requirement for high quality confinement and in such systems only small oscillatory radial banana guiding center excursions from the average drift surface occur. The guiding center radial coordinate is then the leading term for a more precise radial drift invariant I _r , corrected by oscillatory “banana ripple” terms. An analytical expression for the radial invariant is derived for long-thin quadrupolar mirror equilibria. The formula for the invariant is then used in a Vlasov distribution function. Comparisons are first made with Vlasov equilibria using the adiabatic parallel invariant. To model radial density profiles, it is necessary to use the radial invariant (the parallel invariant is insufficient for this). The results are also compared with a fluid approach. In several aspects, the fluid and Vlasov system with the radial invariant give analogous predictions. One difference is that the parallel current associated with finite banana widths could be derived from the radial invariant.     

A Note on Unification of Translational Shape Invariant Potential and Scaling Shape Invariant Potential

This article puts forward a general shape invariant potential, which includes the translational shape invariant potential and scaling shape invariant potential as two particular cases, and derives the set of linear differential equations for obtaining general solutions of the generalized shape invariance condition.     

A Note on Unification of Translational Shape Invariant Potential and Scaling Shape Invariant Potential

This article puts forward a general shape invariant potential, which includes the translational shape invariant potential and scaling shape invariant potential as two particular cases, and derives the set of linear differential equations for obtaining general solutions of the generalized shape invariance condition.     

Invariant circles for the piecewise linear standard map

We investigate invariant circles for a one-parameter family of piecewise linear twist homeomorphisms of the annulus. We show that invariant circles of all types and rotation numbers occur and we classify them into families. We compute parameter ranges in which there are no invariant circles.     

非保守动力学系统Noether对称性的摄动与绝热不变量

基于非保守系统的El-Nabulsi动力学模型, 研究了非保守动力学系统Noether对称性的摄动与绝热不变量问题.首先, 引入El-Nabulsi在分数阶微积分框架下基于Riemann-Liouville分数阶积分提出的类分数阶变分问题, 列出非保守系统的Euler-Lagrange方程; 其次, 给出了Noether准对称变换的定义和判据, 建立了Noether对称性与不变量之间的关系, 得到了精确不变量; 最后, 提出并研究了该系统受小扰动作用后Noether对称性的摄动与绝热不变量问题, 证明了绝热不变量存在的条件及形式, 并举例证明结果的应用. 关键词： 非保守系统 El-Nabulsi动力学模型 对称性摄动 绝热不变量     

Impulse formulations of Hall magnetohydrodynamic equations

Impulse formulations of Hall magnetohydrodynamic (MHD) equations are developed. The Lagrange invariance of a generalized ion magnetic helicity is established for Hall MHD. The physical implications of this Lagrange invariant are discussed. The discussion is then extended to compressible Hall MHD and a generalized ion magnetic potential helicity Lagrange invariant is established. The physical implications of the generalized ion magnetic potential helicity Lagrange invariant are shown to be the same, as to be expected, as those of the generalized ion magnetic helicity Lagrange invariant.     

单面约束Birkhoff系统对称性的摄动与绝热不变量

研究了具有单面约束的Birkhoff系统在小扰动作用下对称性的摄动与不变量之间的关系.首先,建立了系统的运动微分方程,给出了系统的精确不变量;其次,基于力学系统的高阶绝热不变量的概念,得到了系统的绝热不变量存在的条件及其形式;最后,举例说明结果的应用 关键词： 分析力学 单面约束 Birkhoff系统 对称性 摄动 绝热不变量     

Eta invariants,charge fractionalization and anomalies

Starting with the observation that the fermionic number operator of a charge fractionalization problem is the (inverse) Mellin transform of the eta invariant of Atiyah, Patodi and Singer, we calculate the eta invariant for odd-dimensional unit spheres. We obtain an interesting interpretation for both the Goldstone-Wilczek mass relation and the eta invariant (in this case, a generalization of the classical conformal invariant of Blaschke et al.). We briefly discuss the possible anomalies of the implied theory by connecting the eta invariant with the spin-index theorem. We show that the known relationship between Abelian and nonabelian anomalies is an interplay between the spin-index theorem and Bott periodicity. Some related questions are also discussed.