守恒矢量和不变动量的应用

Runge-Lenz矢量和角动量是平方反比有心力场中的两个守恒矢量。介绍了利用守恒矢量简单确定不变动量和旋转不变动量的方法，并利用不变动量和旋转不变动量得到了适合引力和斥力的动量、速度、轨道方程和能量的计算公式，同时也给出了守恒量、不变量与轨道几何参数的关系式。

Exact Invariants and Adiabatic Invariants of Nonholonomic Variable Mass Systems

By the theory of symmetries and conserved quantities, the exact invariants and adiabatic invariants of nonholonomic variable mass systems are studied. The perturbation problem of symmetries for the nonholonomic variable mass systems under small excitation is discussed. The concept of high order adiabatic invariant is presented, and the form of exact invariants and adiabatic invariants as well as the conditions for their existence are given. Then the corresponding inverse problem is studied.

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian Systems

For a relativistic Birkhoffian system, the Lie symmetrical perturbation and adiabatic invariants of generalized Bojman type are studied under general infinitesimal transformations. On the basis of the invariance of relativistic Birkhotfian equations under general infinitesimal transformations,Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The exact invariants in the form of generalized Hojman conserved quantities led by the Lie symmetries of relativistic Birkhoffian system without perturbations are given. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for relativistic Birkhoffian system with the action of small disturbance is investigated, and a new type of adiabatic invariants of the system is obtained. In the end of the paper, an example is given to illustrate the application of the results.

Perturbation of Symmetries and Hojman Adiabatic Invariants for Mechanical Systems with Unilateral Holonomic Constraints

The perturbation of symmetries and adiabatic invariants for mechanical systems with unilateral holonomic constraints are studied. The exact invariant in the form of Hojman led by special Lie symmetries for an undisturbed system with unilateral constraints is given. Based on the concept of high-order adiabatic invariant of mechanical systems, the perturbation of Lie symmetries for the system under the action of small disturbance is investigated, and a new adiabatic invariant for the system with unilateral holonomic constraints is obtained, which can be called Hojman adiabatic invariant. In the end of the paper, an example is given to illustrate the application of the results.

Perturbation and Adiabatic Invariants of Mei Symmetry for Nonholonomic Mechanical Systems

Based on the concept of adiabatic invariant, the perturbation and adiabatic invariants of the Mei symmetry for nonholonomic mechanical systems are studied. The exact invariants of the Mei symmetry for the system without perturbation are given. The perturbation to the Mei symmetry is discussed and the adiabatic invariants of the Mei symmetry for the perturbed system are obtained.

Perturbation to Noether-Mei Symmetry and Adiabatic Invariants for Nonholonomic Mechanical Systems in Phase Space

Based on the concept of adiabatic invariant, the perturbation to Noether Mei symmetry and adiabatic invariants for nonholonomie mechanical systems in phase space are studied. The definition of the perturbation to Noether-Mei symmetry for the system is presented, and the criterion of the perturbation to Noether-Mei symmetry is given. Meanwhile, the Noether adiabatic invariants and the Mei adiabatic invariants for the perturbed system are obtained.

Human action recognition based on chaotic invariants

A new human action recognition approach was presented based on chaotic invariants and relevance vector machines(RVM).The trajectories of reference joints estimated by skeleton graph matching were adopted for representing the nonlinear dynamical system of human action.The C-C method was used for estimating delay time and embedding dimension of a phase space which was reconstructed by each trajectory.Then,some chaotic invariants representing action can be captured in the reconstructed phase space.Finally,RVM was used to recognize action.Experiments were performed on the KTH,Weizmann and Ballet human action datasets to test and evaluate the proposed method.The experiment results show that the average recognition accuracy is over91.2%,which validates its effectiveness.

Perturbation to Unified Symmetry and Adiabatic Invariants for Relativistic Hamilton Systems

Based on the concept of adiabatic invariant, the perturbation to unified symmetry and adiabatic invariants for relativistic Hamilton systems are studied. The definition of the perturbation to unified symmetry for the system is presented, and the criterion of the perturbation to unified symmetry is given. Meanwhile, the Noether adiabatic invariants, the generalized Hojman adiabatic invariants, and the Mei adiabatic invariants for the perturbed system are obtained.

Mei Adiabatic Invariants Induced by Perturbation of Mei Symmetry for Nonholonomic Controllable Mechanical Systems

Two types of Mei adiabatic invariants induced by perturbation of Mei symmetry for nonholonomic controllablemechanical systems are reported.Criterion and restriction equations determining Mei symmetry after beingdisturbed of the system are established.Form and existence condition of Mei adiabatic invariants are obtained.

Perturbation to Lie-Mei Symmetry and Adiabatic Invariants for Birkhoffian Systems

Based on the concept of adiabatic invariant,the perturbation to Lie-Mei symmetry and adiabatic invariantsfor Birkhoffian systems are studied.The definition of the perturbation to Lie-Mei symmetry for the system is presented,and the criterion of the perturbation to Lie-Mei symmetry is given.Meanwhile,the Hojman adiabatic invariants andthe Mei adiabatic invariants for the perturbed system are obtained.

Fast image matching algorithm based on affine invariants

Feature-based image matching algorithms play an indispensable role in automatic target recognition （ATR）. In this work, a fast image matching algorithm （FIMA） is proposed which utilizes the geometry feature of extended centroid （EC） to build affine invariants. Based on at-fine invariants of the length ratio of two parallel line segments, FIMA overcomes the invalidation problem of the state-of-the-art algorithms based on affine geometry features, and increases the feature diversity of different targets, thus reducing misjudgment rate during recognizing targets. However, it is found that FIMA suffers from the parallelogram contour problem and the coincidence invalidation. An advanced FIMA is designed to cope with these problems. Experiments prove that the proposed algorithms have better robustness for Gaussian noise, gray-scale change, contrast change, illumination and small three-dimensional rotation. Compared with the latest fast image matching algorithms based on geometry features, FIMA reaches the speedup of approximate 1.75 times. Thus, FIMA would be more suitable for actual ATR applications.

Dispersion Energy in a Molecule Composed of Two Anisotropic Vibrating Dipoles

For the first time, we derive the dispersion energy for a molecule which involves the anisotropic dipole interaction by virtue of the invariant eigen-operator method, which greatly simplifies the usual calculation if one uses the Schroedinger equation.

Perturbation to Mei Symmetry and Adiabatic Invariants for Nonholonomic Systems in Terms of Quasi-Coordinates

Abstract Based on the concept of adiabatic invariant, the perturbation to Mei symmetry and adiabatic invariants for nonholonomic mechanical systems in terms of quasi-coordinates are studied. The definition of the perturbation to Mei symmetry for the system is presented, and the criterion of the perturbation to Mei symmetry is given. Meanwhile, the Mei adiabatic invariants for the perturbed system are obtained.

A New Type of Mei Adiabatic Invariant Induced by Perturbation to Mei Symmetry of Hamiltonian Systems

Considering full perturbation to infinitesimal generators in the Mei structure equation,a new type of Meiadiabatic invariant induced by perturbation to Mei symmetry for Hamiltonian system was reported.

Invariance principles for the law of the iterated logarithm under G-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm(LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen's invariance principle to the case where probability measure is no longer additive. Furthermore,we give some examples as applications.

Generating Semi-Algebraic Invariants for Non-Autonomous Polynomial Hybrid Systems

Hybrid systems are dynamical systems with interacting discrete computation and continuous physical processes, which have become more common, more indispensable, and more complicated in our modern life. Particularly, many of them are safety-critical, and therefore are required to meet a critical safety standard. Invariant generation plays a central role in the verification and synthesis of hybrid systems. In the previous work, the fourth author and his coauthors gave a necessary and sufficient condition for a semi-algebraic set being an invariant of a polynomial autonomous dynamical system, which gave a confirmative answer to the open problem. In addition, based on which a complete algorithm for generating all semi-algebraic invariants of a given polynomial autonomous hybrid system with the given shape was proposed. This paper considers how to extend their work to non-autonomous dynamical and hybrid systems. Non-autonomous dynamical and hybrid systems are with inputs, which are very common in practice; in contrast, autonomous ones are without inputs. Furthermore, the authors present a sound and complete algorithm to verify semi-algebraic invariants for non-autonomous polynomial hybrid systems. Based on which, the authors propose a sound and complete algorithm to generate all invariants with a pre-defined template.

On the hyperbolicity of flows

Let X be a C1 vector field on a compact boundaryless Riemannian manifold M（dim M≥2）,and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = Es Eu with Es uniformly contracting and Eu uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.

The form-invariance of wave equations without requiring a priori relations between field variables

According to the principle of relativity,the equations describing the laws of physics should have the same forms in all admissible frames of reference,i.e.,form-invariance is an intrinsic property of correct wave equations.However,so far in the design of metamaterials by transformation methods,the form-invariance is always proved by using certain relations between field variables before and after coordinate transformation.The main contribution of this paper is to give general proofs of form-invariance of electromagnetic,sound and elastic wave equations in the global Cartesian coordinate system without using any assumption of the relation between field variables.The results show that electromagnetic wave equations and sound wave equations are intrinsically form-invariant,but traditional elastodynamic equations are not.As a by-product,one can naturally obtain new elastodynamic equations in the time domain that are locally accurate to describe the elastic wave propagation in inhomogeneous media.The validity of these new equations is demonstrated by some numerical simulations of a perfect elastic wave rotator and an approximate elastic wave cloak.These findings are important for solving inverse scattering problems in many fields such as seismology,nondestructive evaluation and metamaterials.