含哈密顿算子的并联式内积张量泛函变分问题
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摘要
讨论变分法中含哈密顿算子即梯度、散度和旋度的并联式张量的泛函变分问题.根据n阶张量并联式内积和串联式内积运算规则,给出张量泛函变分问题的基本引理.提出并证明含哈密顿算子的张量泛函变分问题的定理;通过直接对张量的梯度、散度和旋度进行变分,得到欧拉方程和相应的自然边界条件.通过若干算例验证了欧拉方程的正确性.扩展伴随算子的内涵,提出右伴随算子的概念,讨论伴随算子或自伴算子与梯度、散度和旋度算子的关系,指出所讨论的泛函变分问题实质上是符合伴随算子或自伴算子定义的运算.
The variational problems of the functional depending on parallel-type inner product tensor in the calculus of variations were discussed, the tensor in the functional can include Hamiltonian operator namely the gradient, divergence and curl operator. According to the parallel-type and serial-type inner product operation rules of the n-th order tensor, the fundamental lemma of the variational operations of the functional depending on the tensor was given. The theorem of the variational problem of the functional depending on the tensor with Hamiltonian operators was proposed and proved; Through making directly variation to the tensors with Hamiltonian operators, the Euler equation and the corresponding natural boundary conditions were obtained. The correctness of the Euler equation was verified with some functional examples. The connotation of the dajoint operator was extended, the conception of the right adjoint operator was proposed, the relationships between the adjoint operator or self-adjoint operator and the gradient, divergence and curl operator were discussed, it is pointed out that the variational problem of the functional discussed is essentially operation conforming to the definition of adjoint operator operation or self-adjoint operator.
The variational problems of the functional depending on parallel-type inner product tensor in the calculus of variations were discussed, the tensor in the functional can include Hamiltonian operator namely the gradient, divergence and curl operator. According to the parallel-type and serial-type inner product operation rules of the n-th order tensor, the fundamental lemma of the variational operations of the functional depending on the tensor was given. The theorem of the variational problem of the functional depending on the tensor with Hamiltonian operators was proposed and proved; Through making directly variation to the tensors with Hamiltonian operators, the Euler equation and the corresponding natural boundary conditions were obtained. The correctness of the Euler equation was verified with some functional examples. The connotation of the dajoint operator was extended, the conception of the right adjoint operator was proposed, the relationships between the adjoint operator or self-adjoint operator and the gradient, divergence and curl operator were discussed, it is pointed out that the variational problem of the functional discussed is essentially operation conforming to the definition of adjoint operator operation or self-adjoint operator.
引文
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[17] Pan Xingbin,Qi Yuanwei.Asymptotics of minimizers of variational problems involving curl functional[J].Journal of Mathematical Physics,2000,41(7):5033-5063.
[18] 梁立孚,周平.Lagrange 方程应用于流体动力学[J].哈尔滨工程大学学报,2018,39(1):33-39.Liang Lifu,Zhou Ping.Application of Lagrange equation in fluid mechanics[J].Journal of Harbin Engineering University,2018,39(1):33-39.(in Chinese)
[2] Merches I.Variational principle in magnetohydrodynamics[J].The Physics of Fluids,1969,12(10):2225-2227.
[3] Kotiuga P R.Variational principles for three-dimensional magnetostatics based on helicity[J].J Appl Phys,1988,63(8):3360-3362.
[4] Brownstein K R.Quantum mechanical momentum operators and hamilton’s variational principle[J].American Journal of Physics,1976,44(7):677-679.
[5] Goldstein H.Classical mechanics [M].2nd ed.[S.l.]:Addison-Wesley Publishing Company,1980.
[6] Guo Zhongheng.Unified theory of variation principles in non-linear theory of elasticity[J].Applied Mathematics and Mechanics:English Edition,1980,I(1):1-22.
[7] 钱伟长.弹性理论中各种变分原理的分类[J].应用数学和力学,1984,5(5):765-770.Chien Weizang.Classification of variational principles in elasticity[J].Applied Mathematics and Mechanics,1984,5(5):765-770.(in Chinese)
[8] 钱仲范,刁颖敏.人工心瓣最佳膜面形状的变分法[J].工程数学学报,1989,6(1):123-126.Qian Zhongfan,Diao Yingmin.The variational methed for the optimal membrane shape of the cardiac valve prostheses[J].Journal of Engineering Mathematics,1989,6(1):123-126.(in Chinese)
[9] Freiman M,Joskowicz L,Sosna J.A variational method for vessels segmentation:algorithm and application to liver vessels visualization[C]//Proceedings of SPIE.[S.l.]:SPIE,2009:1-8.
[10] Rudin L I,Osher S,Fatemi E.Nonlinear total variation based noise removal algorithms[J].Physica D,1992,60:259-268.
[11] Xu Chenyang,Prince J L.Gradient vector flow:a new external force for snakes[C]//Proceeding of IEEE Conf Computer Vision and Pattern Recognition.[S.l.]:IEEE,1997:66-71.
[12] 李继先.自适应网格生成技术及其应用[J].武汉水运工程学院学报,1992,16(1):73-81.Li Jixian.Adaptive mesh generation and its application[J].Journal of Wuhan University Water Transportation Engineering,1992,16(1):73-81.(in Chinese)
[13] 张花艳.基于变分方法的网格处理[D].合肥:中国科学技术大学,2015.Zhang Huayan.Mesh processing based on variational approaches[D].Hefei:University of Science and Technology of China,2015.(in Chinese)
[14] Spirn D.Vortex motion law for the Schr?dinger-Ginzburg-Landau equations[J].Siam J Math Anal,Society for Industrial and Applied Mathematics,2003,34(6):1435-1476.
[15] 潘兴斌.超导与液晶边界层现象的数学问题[J].中国科学:数学,2018,48(1):83-110.Pan Xingbin.Mathematical problems of boundary layer behavior of superconductivity and liquid crystals[J].Scientia Sinica Mathematica,2018,48(1):83-110.(in Chinese)
[16] 苏景辉.用微分算子表示的欧拉方程[J].哈尔滨船舶工程学院学报,1991,12(1):134-139.Su Jinghui.Euler equation with differential operator .Journal of Harbin Shipbuilding Engineering Institute,1991,12(1):134-139.(in Chinese)
[17] Pan Xingbin,Qi Yuanwei.Asymptotics of minimizers of variational problems involving curl functional[J].Journal of Mathematical Physics,2000,41(7):5033-5063.
[18] 梁立孚,周平.Lagrange 方程应用于流体动力学[J].哈尔滨工程大学学报,2018,39(1):33-39.Liang Lifu,Zhou Ping.Application of Lagrange equation in fluid mechanics[J].Journal of Harbin Engineering University,2018,39(1):33-39.(in Chinese)