HAMILTONIAN MECHANICS ON KAHLER MANIFOLDS

Using the mechanical principle, the theory of modern geometry and advanced calculus, Hamiltonian mechanics was generalized to Kahler manifolds, and the Hamiltonian mechanics on Kahler manifolds was established. Then the complex mathematical aspect of Hamiltonian vector field and Hamilton's equations was obtained, and so on.     

分离变量法与哈密尔顿体系

数学物理与力学中用分离变量法求解偏微分方程经常导致自共轭算子的sturmLiouville问题,在此基础上而得以展开求解。然而在应用中有大量问题并不能导致自共轭算子。本文通过最小势能变分原理,选用状态变量及其对偶变量,导向一般变分原理。利用结构力学与最优控制的模拟理论,导向哈密尔顿体系。将有限维的理论推广到相应的哈密尔顿算子矩阵及共轭辛矩阵代数的理论。拓广了经典的分离变量法,证明了全状态本征函数向量的共轭辛正交归一性质及按本征函数向量展开的理论。以条形板为例,说明了应用。     

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).     

平面粘性流体扰动与哈密顿体系

通过变分原理,将哈密顿体系的理论引入到平面粘性流体扰动的问题中,导出一套哈密顿算子矩阵的本征函数向量展开求解问题的方法。基于直接法求解流体力学基本方程,导出流场一般特征关系,通过本征值的求解及本征向量的叠加,得到波扰动解,继可分析流场端部效应。从而在该领域用在哈密顿体系下辛几何空间中研究问题的方法代替了传统在拉格朗日体系欧几里德空间分析问题的方法。为流体力学的研究提供一条新途径。     

Dissipativity and Lie-admissible algebras

Summary The validity of Hamiltonian mechanics and Lie algebra invariance for free fields and their difficulties for interacting fields are recalled. The role of dissipativity is discussed as an approach allowing the introduction for interacting regions of larger analytical dynamics and algebraic formulations related by the enlarged bracket. A working criterion is introduced in terms of Lie-admissible algebras and the pseudo-Hamiltonian mechanics introduced by R. J. Duffin for discrete dissipative systems is considered as an explicit choice able to reduce to the Hamiltonian mechanics when the systems become conservative. An example of dissipative plasma is explicitly investigated. Furthermore the procedure is extended to continuous systems and classical interpolating dissipative fields induced by Lie-admissible structures are constructed.

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Sommario Si richiama la validità della Meccanica Hamiltoniana e l'invarianza secondo le algebre di Lie per campi liberi, insieme alle loro difficoltà per campi interagenti. Si discute il ruolo della dissipatività dei campi interpolati come un indirizzo che permette l'introduzione per le regioni di interazione di più larghe strutture analitiche ed algebriche connesse da parentesi di Poisson generalizzate. Si introduce un criterio di lavoro espresso in termini di algebre Lie-ammissibili, il quale da luogo alla Meccanica Pseudo-Hamiltoniana introdotta da R. J. Duffin nel 1962 per sistemi classici, discreti e dissipativi. Come esempio di applicazione fisica si costruisce un modello di plasma dissipativo e si mostra come i parametri della formulazione influenzano il vettore corrente elettrica ed il tensore di conduttività. Inoltre la procedura viene estesa a sistemi classici, continui e dissipativi costruendo, come applicazione, alcuni esempi di equazioni differenziali per campi interpolati indotte da strutture di tipo Lieammissibile.

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This research was carried out at the Center for Theoretical Studies, University of Miami, Coral Gables, Florida and was supported by U.S. Air Force Contract No. 49 (638)-1260 and NASA Contract No. NASA NGR 10-007-010.     

弹性与断裂力学复变方法研究进展——纪念弹性与断裂力学复变函数法提出100周年

1909年,俄国数学力学家哥洛索夫首先将复变函数方法应用于二维弹性力学问题,揭开了弹性力学复变方法研究的序幕.100年来,复变方法在求解弹性与断裂力学问题中取得了很大发展,特别在断裂力学中的应用尤为成功.2009年恰逢弹性力学复变方法提出100周年,该文试图总结100年来复变方法在经典断裂力学、复合材料断裂力学、新型材料断裂力学以及三维空间断裂问题中的发展与应用,以作纪念.     

On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported

This paper presents a bridging research between a modeling methodology in quantum mechanics/relativity and elasticity. Using the symplectic method commonly applied in quantum mechanics and relativity, a new symplectic elasticity approach is developed for deriving exact analytical solutions to some basic problems in solid mechanics and elasticity which have long been bottlenecks in the history of elasticity. In specific, it is applied to bending of rectangular thin plates where exact solutions are hitherto unavailable. It employs the Hamiltonian principle with Legendre’s transformation. Analytical bending solutions could be obtained by eigenvalue analysis and expansion of eigenfunctions. Here, bending analysis requires the solving of an eigenvalue equation unlike in classical mechanics where eigenvalue analysis is only required in vibration and buckling problems. Furthermore, unlike the semi-inverse approaches in classical plate analysis employed by Timoshenko and others such as Navier’s solution, Levy’s solution, Rayleigh–Ritz method, etc. where a trial deflection function is pre-determined, this new symplectic plate analysis is completely rational without any guess functions and yet it renders exact solutions beyond the scope of applicability of the semi-inverse approaches. In short, the symplectic plate analysis developed in this paper presents a breakthrough in analytical mechanics in which an area previously unaccountable by Timoshenko’s plate theory and the likes has been trespassed. Here, examples for plates with selected boundary conditions are solved and the exact solutions discussed. Comparison with the classical solutions shows excellent agreement. As the derivation of this new approach is fundamental, further research can be conducted not only on other types of boundary conditions, but also for thick plates as well as vibration, buckling, wave propagation, etc.     

DYNAMICS IN NEWTONIAN-RIEMANNIAN SPACE-TIME (Ⅳ)

ＩｎｔｒｏｄｕｃｔｉｏｎＬａｇｒａｎｇｉａｎｍｅｃｈａｎｉｃｓｉｓａｎｉｍｐｏｒｔａｎｔｐａｒｔｏｆａｎａｌｙｔｉｃａｌｍｅｃｈａｎｉｃｓ.ＩｔｉｓａｇｅｎｅｒａｌｉｚａｔｉｏｎｏｆＮｅｗｔｏｎｉａｎｍｅｃｈａｎｉｃｓ.Ｎｅｗｔｏｎｉａｎｍｅｃｈａｎｉｃｓｉｓｅｓｔａｂｌｉｓｈｅｄｏｎｔｈｅｃｏｎｆｉｇｕｒａｔｉｏｎｓｐａｃｅ—Ｒｉｅｍａｎｎｉａｎｍａｎｉｆｏｌｄ .Ｌａｇｒａｎｇｉａｎｍｅｃｈａｎｉｃｓｉｓｅｓｔａｂｌｉｓｈｅｄｏｎｉｔｓｔａｎｇｅｎｔｂｕｎｄｌｅ ,Ｈａｍｉｌｔｏｎｉａｎｍｅｃｈａ…     

DYNAMICS IN NEWTONIAN-RIEMANNIAN SPACE-TIME(Ⅳ)

Lagrangian mechanics in Newtonian-Riemannian space-time and relationship between Lagrangian mechanics and Newtonian mechanics, and between Lagrangian mechanics and Hamiltonian mechanics in N-R space-time are discussed.     

Caputo导数下分数阶Birkhoff系统的准对称性与分数阶Noether定理

应用分数阶模型可以更准确地描述和研究复杂系统的动力学行为和物理过程,同时Birkhoff力学是Hamilton力学的推广,因此研究分数阶Birkhoff系统动力学具有重要意义.分数阶Noether定理揭示了Noether对称变换与分数阶守恒量之间的内在联系,但是当变换拓展为Noether准对称变换时,该定理的推广遇到了很大的困难.本文基于时间重新参数化方法提出并研究Caputo导数下分数阶Birkhoff系统的Noether准对称性与守恒量.首先,将时间重新参数化方法应用于经典Birkhoff系统的Noether准对称性与守恒量研究,建立了相应的Noether定理;其次,基于分数阶Pfaff作用量分别在时间不变的和一般单参数无限小变换群下的不变性给出分数阶Birkhoff系统的Noether准对称变换的定义和判据,基于Frederico和Torres提出的分数阶守恒量定义,利用时间重新参数化方法建立了分数阶Birkhoff系统的Noether定理,从而揭示了分数阶Birkhoff系统的Noether准对称性与分数阶守恒量之间的内在联系.分数阶Birkhoff系统的Noether对称性定理和经典Birkhoff系统的Noether定理是其特例.最后以分数阶Hojman-Urrutia问题为例说明结果的应用.     

复杂微力－电系统的细微尺度力学

现代高新技术的崛起，提出了大量新的经典力学所不能完全包容的力学问题。这将是现代应用力学发展的巨大动力。微电子技术中微电子材料、器件、系统和微电子－机械系统（ｍｉｃｒｏｅｌｅｃｔｒｏ－ｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍ，ＭＥＭＳ）所组成的复杂微力－电系统，是跨世纪发展的新科技方向，本文简要介绍了复杂微力，电系统的工业技术背景和发展；综述了这一领域存在的力学问题，主要讨论细微尺度下的结构力学与破坏力学。并评介与展望了这一新的力学领域的研究状况和发展趋势。     

Hamiltonian systems in elasticity and their variational principles

ＨＡＭＩＬＴＯＮＩＡＮＳＹＳＴＥＭＳＩＮＥＬＡＳＴＩＣＩＴＹＡＮＤＴＨＥＩＲＶＡＲＩＡＴＩＯＮＡＬＰＲＩＮＣＩＰＬＥＳＷａｎｇＺｈｉ－ｇｕｏ（王治国）（ＲｅｓｅａｒｃｈＩｎｓｔｉｔｕｔｅｏｆＶｉｂｒａｔｉｏｎＥｎｇｉｎｅｅｒｉｎｇ,ＮａｎｊｉｎｇＵｎ...     

Low‐Dimensional Lattice Groups for the Continuum Mechanics of Phase Transitions in Crystals

First I review and adapt some classical and recent formulations of problems relating to the classification of two‐ and three‐dimensional lattices. Then I use these methods to calculate lattice groups that are of interest in the continuum mechanics of phase transitions in crystals, and to construct invariant neighbourhoods that are useful in the same context. The paper is designed to provide a perspective of similar calculations which employ methods of traditional crystallography. (Accepted February 9, 1998)     

Symmetry Reduced Dynamics of Charged Molecular Strands

The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids. On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form.     

On `best' shell models – From classical shells, degenerated and multi-layered concepts to 3D

Summary Problems of solid mechanics are most generally formulated within 3D continuum mechanics. However, engineering models favor reduced dimensions, in order to portray mechanical properties by surface or curvilinear approximations. Such attempts for dimensional reduction constitute interactions between theoretical formulations and numerical techniques. A classical reduced model for thin bodies is represented by shell theory, an approximation in terms of resultants and first-order moments. If the shell theory, with its inherent errors, is considered as qualitatively insufficient for a particular problem, a further improvement is given by solid shell models, which are gained by direct linear interpolation of the 3D kinematic relations. They improve considerably the analytic capabilities for shells, especially when their congenital locking effects are handled by variational `convergence tricks'. The next step towards 3D quality are layered shells or solid shell elements. The present paper compares these three approximation stages from the point of view of multi-director (integral) transformations of classical continuum mechanics. It offers physical convergence requirements for each of the treated models. Partial support to the present study by the German Science Foundation (DFG) within the Special Research Center (SFB) 398 is gratefully acknowledged.     

非线性水波Hamilton系统理论与应用研究进展

概述了辛几何理论与辛算法在Hamilton力学中的应用，综述非线性水波的Hamilton理论研究进展．阐述非线性水波Hamilton变分原理与方法的优越性与局限性，探讨KdV方程和BBM方程的Hamilton描述、对称性与守恒律，提出非线性水波Hamilton描述研究中有待进一步研究的问题和解法设想．     

复杂微力－电系统的细微尺度力学

现代高新技术的崛起,提出了大量新的经典力学所不能完全包容的力学问题。这将是现代应用力学发展的巨大动力。微电子技术中微电子材料、器件、系统和微电子－机械系统（ｍｉｃｒｏｅｌｅｃｔｒｏ－ｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍ,ＭＥＭＳ）所组成的复杂微力－电系统,是跨世纪发展的新科技方向,本文简要介绍了复杂微力,电系统的工业技术背景和发展;综述了这一领域存在的力学问题,主要讨论细微尺度下的结构力学与破坏力学。并评介与展望了这一新的力学领域的研究状况和发展趋势。      

一类偏微分方程的Hamilton正则表示

主要给出一系列关于力学中的偏微分方程的无穷维Ｈａｍｉｌｔｏｎ正则表示．其中包括变系数线性偏微分方程,ＫｄＶ方程,ＭＫｄＶ方程,ＫＰ方程,Ｂｏｕｓｉｎｅｓｑ方程等的无穷维Ｈａｍｉｌｔｏｎ正则表示．     

Monodromy and nonintegrability in complex Hamiltonian systems

This paper investigates the monodromy representation of the normal variational equation along a phase curve of a two-dimensional complex analytic Hamiltonian system. Geometrical conditions are presented which guarantee reducibility, together with additional hypotheses to ensure complete reducibility. Symmetries in the equations are treated in detail. Applications to establishing the nonintegrability of specific systems are presented.     

HAMILTONIAN SYSTEM AND SIMPLETIC GEOMETRY IN MECHANICS OF COMPOSITE MATERIALS (Ⅰ)——FUNDAMENTAL THEORY

For the first time, Hamiltonian system used in dynamics is introduced to formulate statics and Hamdtonian equation is derived corresponding to the original governing equation, which enables separation of variables to work and eigen function to be obtained for the boundary problem. Consequently, analytical and semi-analytical solutions can be got. The method is especially suitable to solve rectangular plane problem and spatial prism in elastic mechanics. The paper presents a new idea to solve partially differential equation in solid mechanics. The flexural problem and phane stress problem of laminated plate are studied in detail.