Lagrangian reduction of generalized nonholonomic systems

In this paper we study the Lagrangian reduction of generalized nonholonomic systems (GNHS) with symmetry. We restrict ourselves to those GNHS, defined on a configuration space Q$Q$, with kinematic constraints given by a general submanifold _CK⊂TQ$C_K\subset TQ$, and variational constraints given by a distribution _CV$C_V$ on Q$Q$. We develop a reduction procedure that is similar to that for nonholonomic systems satisfying d’Alembert’s principle, i.e. with _CK$C_K$ a distribution and _CV=_CK$C_V=C_K$. Special care is taken in identifying the geometrical structures and mappings involved. We illustrate the general theory with an example.     

基于热质理论的Hamilton原理

基于热质理论,类比经典力学,给出了热质运动遵循的Hamilton原理以及相应的导热Lagrange方程.由于考虑了热质动能,热质运动的Hamilton原理有望应用于非Fourier效应的讨论,在忽略热质动能时,回归到Fourier热学.应用Lagrange方程对含内热源一维瞬态导热问题进行了近似求解,计算结果与解析解符合较好.从分析力学的角度对传热理论以及热学与力学的统一做了新的阐释,指出了现有文献中采用分析力学方法讨论导热问题时存在的某些不足,为导热问题的近似求解提供了新的思路,同时也说明了热质和热质能     

Stochastic least-action principle for the incompressible Navier-Stokes equation

We formulate a stochastic least-action principle for solutions of the incompressible Navier-Stokes equation, which formally reduces to Hamilton’s principle for the incompressible Euler solutions in the case of zero viscosity. We use this principle to give a new derivation of a stochastic Kelvin Theorem for the Navier-Stokes equation, recently established by Constantin and Iyer, which shows that this stochastic conservation law arises from particle-relabelling symmetry of the action. We discuss issues of irreversibility, energy dissipation, and the inviscid limit of Navier-Stokes solutions in the framework of the stochastic variational principle. In particular, we discuss the connection of the stochastic Kelvin Theorem with our previous “martingale hypothesis” for fluid circulations in turbulent solutions of the incompressible Euler equations.     

Detecting coherent structures using braids

The detection of coherent structures is an important problem in fluid dynamics, particularly in geophysical applications. For instance, knowledge of how regions of fluid are isolated from each other allows prediction of the ultimate fate of oil spills. Existing methods detect Lagrangian coherent structures, which are barriers to transport, by examining the stretching field as given by finite-time Lyapunov exponents. These methods are very effective when the velocity field is well-determined, but in many applications only a small number of flow trajectories are known, for example when dealing with oceanic float data. We introduce a topological method for detecting invariant regions based on a small set of trajectories. In this method, we regard the two-dimensional trajectory data as a braid in three dimensions, with time being the third coordinate. Invariant regions then correspond to trajectories that travel together and do not entangle other trajectories. We detect these regions by examining the growth of hypothetical loops surrounding sets of trajectories, and searching for loops that show negligible growth.     

Discrete Lagrangian Reduction,Discrete Euler–Poincaré Equations,and Semidirect Products

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincaré equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.     

Routh Order Reduction Method of Relativistic Birkhoffian Systems

Routh order reduction method of the relativistic Birkhoffian equations is studied.For a relativistic Birkhoffian system,the cyclic integrals can be found by using the perfect differential method.Through these cyclic integrals,the order of the system can be reduced.If the relativistic Birkhoffian system has a cyclic integral,then the Birkhoffian equations can be reduced at least by two degrees and the Birkhoffian form can be kept.The relations among the relativistic Birkhoffian mechanics,the relativistic Hamiltonian mechanics,and the relativistic Lagrangian mechanics are discussed,and the Routh order reduction method of the relativistic Lagrangian system is obtained.And an example is given to illustrate the application of the result.     

Whittaker Order Reduction Method of Relativistic Birkhoffian Systems

The order reduction method of the relativistic Birkhoffian equations is studied. For a relativistic autonomous Birkhoffian system, if the conservative law of the Birkhoffian holds, the conservative quantity can be called the generalized energy integral. Through the generalized energy integral, the order of the system can be reduced. If the relativistic Birkhoffian system has a generalized energy integral, then the Birkhoffian equations can be reduced by at least two degrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics and the relativistic Lagrangian mechanics are discussed, and the Whittaker order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of the result.     

Pin structures and the modified Dirac operator

General theorems on pin structures on products of manifolds and on homogeneous (pseudo-) Riemannian spaces are given and used to find explicitly all such structures on odd-dimensional real projective quadrics, which are known to be non-orientable (Cahen et al. 1993). It is shown that the product of two manifolds has a pin structure if and only if both are pin and at least one of them is orientable. This general result is illustrated by the example of the product of two real projective planes. It is shown how the Dirac operator should be modified to make it equivariant with respect to the twisted adjoint action of the Pin group. A simple formula is derived for the spectrum of the Dirac operator on the product of two pin manifolds, one of which is orientable, in terms of the eigenvalues of the Dirac operators on the factor spaces.     

Discrete variational principle and first integrals for Lagrange--Maxwell mechanico-electrical systems

This paper presents a discrete variational principle and a method to build first-integrals for finite dimensional Lagrange--Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. The discrete variational principle and the corresponding Euler--Lagrange equations are derived from a discrete action associated to these systems. The first-integrals are obtained by introducing the infinitesimal transformation with respect to the generalized coordinates and electric quantities of the systems. This work also extends discrete Noether symmetries to mechanico-electrical dynamical systems. A practical example is presented to illustrate the results.     

Whittaker Order Reduction Method of Relativistic Birkhoffian Systems

The order reduction method of the relativistic Birkhollian equations is studied. For a relativistic autonomous Birkhotffian system, if the conservative law of the Birkhotffian holds, the conservative quantity can be called the generalized energy integral. Through the generalized energy integral, the order of the system can be reduced. If the relativisticBirkhoffian system has a generalized energy integral, then the Birkhoffian equations can be reduced by at least twodegrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics and the relativistic Lagrangian mechanics are discussed, and the Whittaker order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of theresult.     

Singular reduction of implicit Hamiltonian systems

This paper develops the theory of singular reduction for implicit Hamiltonian systems admitting a symmetry Lie group. The reduction is performed at a singular value of the momentum map. This leads to a singular reduced topological space which is not a smooth manifold. A topological Dirac structure on this space is defined in terms of a generalized Poisson bracket and a vector space of derivations, both being defined on a set of smooth functions. A corresponding Hamiltonian formalism is described. It is shown that solutions of the original system descend to solutions of the reduced system. Finally, if the generalized Poisson bracket is nondegenerate, then the singular reduced space can be decomposed into a set of smooth manifolds called pieces. The singular reduced system restricts to a regular reduced implicit Hamiltonian system on each of these pieces. The results in this paper naturally extend the singular reduction theory as previously developed for symplectic or Poisson Hamiltonian systems.     

Two-spinor Formulation of First-Order Gravity Coupled to Dirac Fields

A two-spinor formalism for the Einstein Lagrangian is developed. The gravitational field is regarded as a composite object derived from soldering forms. Our formalism is geometrically and globally well-defined and may be used in virtually any 4m-dimensional manifold with arbitrary signature as well as without any stringent topological requirement on space-time, such as parallelizability. Interactions and feedbacks between gravity and spinor fields are considered. As is well known, the Hilbert–Einstein Lagrangian is second order also when expressed in terms of soldering forms. A covariant splitting is then analysed leading to a first-order Lagrangian which is recognized to play a fundamental role in the theory of conserved quantities. The splitting and thence the first-order Lagrangian depend on a reference spin connection which is physically interpreted as setting the zero level for conserved quantities. A complete and detailed treatment of conserved quantities is then presented.     

Equivalence of Dirac quantization and Schwinger's action principle quantization

We show that the method of Dirac quantization is equivalent to Schwinger's action principle quantization. The relation between the Lagrange undetermined multipliers in Schwinger's method and Dirac's constraint bracket matrix is established and it is explicitly shown that the two methods yield identical (anti)commutators. This is demonstrated in the non-trivial example of supersymmetric quantum mechanics in superspace.     

Dynamical Analysis of the Dow Jones Index Using Dimensionality Reduction and Visualization

Time-series generated by complex systems (CS) are often characterized by phenomena such as chaoticity, fractality and memory effects, which pose difficulties in their analysis. The paper explores the dynamics of multidimensional data generated by a CS. The Dow Jones Industrial Average (DJIA) index is selected as a test-bed. The DJIA time-series is normalized and segmented into several time window vectors. These vectors are treated as objects that characterize the DJIA dynamical behavior. The objects are then compared by means of different distances to generate proper inputs to dimensionality reduction and information visualization algorithms. These computational techniques produce meaningful representations of the original dataset according to the (dis)similarities between the objects. The time is displayed as a parametric variable and the non-locality can be visualized by the corresponding evolution of points and the formation of clusters. The generated portraits reveal a complex nature, which is further analyzed in terms of the emerging patterns. The results show that the adoption of dimensionality reduction and visualization tools for processing complex data is a key modeling option with the current computational resources.     

Optic eikonal,Fermat’s principle and the least action principle

A generalized refractive index in the form of optic eikonal is defined through comparing frame definitions of left-handed and right-handed sets and indicates the sign of the refractive index covered by the quadratic form of the eikonal equation. Fermat’s principle is generalized, and the general refractive law is derived directly. Under this definition, the comparison between Fermat’s principle and the least action principle is made through employing path integral and analogizing L. de Broglie’s theory. Supported by the National Natural Science Foundation of China (Grant No. 60601028)     

Reduced Dirac equation and Lamb shift as off-mass-shell effect in quantum electrodynamics

Based on the accurate experimental data of energy-level differences in hydrogen-like atoms, especially the 1S--2S transitions of hydrogen and deuterium, the necessity of introducing a reduced Dirac equation with reduced mass as the substitution of original electron mass is stressed. Based on new cognition about the essence of special relativity, we provide a reasonable argument for the reduced Dirac equation to have two symmetries, the invariance under the (newly defined) space--time inversion and that under the pure space inversion, in a noninertial frame. By using the reduced Dirac equation and within the framework of quantum electrodynamics in covariant form, the Lamb shift can be evaluated (at one-loop level) as the radiative correction on a bound electron staying in an off-mass-shell state---a new approach eliminating the infrared divergence. Hence the whole calculation, though with limited accuracy, is simplified, getting rid of all divergences and free of ambiguity.     

Hf-N体系的晶体结构预测和性质的第一性原理研究

为了寻找具有优异力学性能的新型超高温陶瓷材料, 结合进化算法和第一性原理, 系统研究了Hf-N二元体系所有稳定存在的化合物及其晶体结构. 除了实验已知的岩盐结构的HfN之外, 本文还找到了Hf₆N(R-3), Hf₃N(P6₃22), Hf₃N₂(R-3m), Hf₅N₆(C2/m)和Hf₃N₄(C2/m)五种新结构, 基于准简谐近似原理计算了这些稳定结构的声子谱以验证其动力学稳定性, 常温甚至更高温度下的吉布斯自由能以验证其高温热力学稳定性. 结果表明, 这些结构是动力学稳定的, 且在1500 K以下都是热力学稳定的. 同时, 本文还列出了在搜索过程中出现的空间对称性较高、能量较低的亚稳态结构, 包括Hf₂N(P4₂/mnm), Hf₄N₃(C2/m), Hf₆N₅(C2/m), Hf₄N₅(I4/m), Hf₃N₄(I-43d)和Hf₃N₄(Pnma). 之后计算了上述所有结构的力学性质(弹性常数、体模量、 剪切模量、 杨氏模量、硬度), 随着N 所占比例的增加, 硬度呈现的整体趋势是先增大后下降, 在Hf₅N₆处取得最大值, 为21 GPa. 其中Hf₃N₂和Hf₄N₅也展现出了较高的硬度, 都为19 GPa. 最后, 计算了这些结构的电子态密度和晶体轨道汉密尔顿分布, 从电子结构的角度分析了力学性能的成因. 研究结果显示, 较强的Hf-N共价键和较低的结构空位率是Hf₅N₆具有优异力学性能的主要原因.     

Large deviation principle for spatial density of local observables from Gibbs distributions

We establish the large deviation principle characterising, in the thermodynamic limit, the exponential decay rates for the probabilities of macroscopic fluctuations of spatial densities generated by local observables from Gibbs lattice systems with absolutely summable interactions.     

ADM electrons and the equivalence principle

Noting that the general relativistic ADM equation for the mass of a sphere of charged dust (with no angular momentum) reveals that the masses of point-like particles are determined solely by their electrical charge, electron models based on extended spheres of such purely electrical dust are examined. It is shown that for all realistic electron models of this type (where the observed electron mass is positive and many orders of magnitude smaller than either the Planck or ADM mass) the electron's bare active gravitational mass must be taken to be negative. Because of the negativity of the bare active gravitational mass, one of the two realistic models leads to a violation of the weak equivalence principle, but the other does not. A means of testing whether negative mass obeys the equivalence principle is mentioned.