论二阶Euler—Lagrange方程的形式和结构

本文讨论m个变元情形,得到结论:m≥2时,二阶方程为Euler-Lagrange方程的充要条件是它为Monge-Ampere方程,其系数满足某些条件。     

Theoretical Maxwelrs Equations, Gauge Field and Their Universality Based on One Conservation Law

The notion of the inner product of vectors is extended to tensors of different orders, which may replace the vector product usually. The essences of the differential and the codiffcrential forms are pointed out： they represent the tangent surface and the normal surface fluxes of a tensor, reslpetivcly. The definitions of the divergence and the curl of a 2D surface flux of a tensor arc obtained. Maxwell＇s equations, namely, the constraction law of field, which were usually established based on two conservation laws of electric charge and imaginary magnetic charge, are derived by the author only by using one conservation law （ mass or fluid flux quantity and so on） and the feature of central field （or its composition）. By the feature of central field （or its composition）, the curl of 2D flux is zero. Both universality of gauge field and the difficulty of magnetic monopole theory （ a magnetic monopole has no effect on electric current just like a couple hasing no effect on the sum of forces） axe presented： magnetic monopole has no the feature of magnet. Finally it is pointed out that the base of relation of mass and energy is already involved in Maxwell＇s equations.     

解变分逆问题的一个方法

在有限元法使用中,物理、力学中都存在着将方程问题转化为变分平稳值这一课题。本文就二阶非线性偏微分方程,在乘一个所谓“积分因子”的条件下,找出这种能转化的必要充分条件,并给出实现这种转化的具体方法,不依賴泛函分析知识。不受阶数限制,有普遍意义。本文特别详细讨论了二维情形,作为这种方法有效的例证,将流体力学的一个非线性方程,转化为变分问题。     

梯度、旋度概念在泛函分析中的表现

本文首先建立“在一点变分”的概念,指出在这方面通常提法的不当;采取视泛函为无穷多个变元的观点。以此,在泛函分析中,较通常文献更自然地引进梯度概念,从而直观简明地阐述了Вайнберг的Euler算子(即梯度算子)判别准则,使其成为几乎不证自明的事实。在深入讨论了梯度概念后,将此判别法推广到包括自然边界条件上。也讨论旋度概念,给出算子分解定理的较一般形式。对照初等微积分,形成泛函分析中二型线积分概念,也就排除了在解变分学逆问题中,构造泛函上现存的偶然性痕迹。最后,对泛函分析中场论作些设想,并导出了一般导算子的判别准则。     

Theoretical Maxwell's Equations, Gauge Field and Their Universality Based on One Conservation Law

Introduction Ampere,Faradayandotherphysicistsproposeda numberofexperimentallawsaboutelectromagnetism.OnthesebasesMaxwellestablishedMaxwell sequa tionsin1862.Meanwhile,accordingtomathematical theoreticalpointofview(consistencyofequations),hesupplementedelectromagnetismwithelectricdis placementvectors.Einsteinpointedoutthat,justaswhatthe ancientshadsaid,peoplemightgraspactualitybypure mathematicalthought.Butintheestablishmentof Maxwell sequations,mathematiciansplayedonlypar tiallyimportantrole…