圆形薄膜振动方程基于Hamilton体系的分离变量法
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摘要
对极坐标系下的振动方程,首先引入合适的对偶变量将其化为Hamilton系统,再结合Bessel函数及双Fourier级数的性质证明了导出的Hamilton算子矩阵的本征函数系的完备性,最后利用展开定理给出了Hamilton系统的解.
For the vibrations of a circular membrane in the polar coordinates, the Hamiltonian system is firstly derived by choosing the appropriate variables. Moreover, the completeness theorem of the eigenfunction system for the Hamiltonian operator is proved by Bessel function and double Fourier series. Finally, the solution of the Hamiltonian system is given by the expansion theorem.
For the vibrations of a circular membrane in the polar coordinates, the Hamiltonian system is firstly derived by choosing the appropriate variables. Moreover, the completeness theorem of the eigenfunction system for the Hamiltonian operator is proved by Bessel function and double Fourier series. Finally, the solution of the Hamiltonian system is given by the expansion theorem.
引文
[1]阿诺尔德著,齐民友译.经典力学的数学方法(第4版)[M].北京:高等教育出版社,2006.
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[3]钟万勰.弹性力学求解新体系[M].大连:大连理工大学出版社,1995.
[4]吴德玉,阿拉坦仓.无穷维Hamilton算子特征函数系的Cauchy主值意义下的完备性[J].中国科学:数学,2008, 38(8):904-912.
[5] Hou G L, Alatancang. Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supposed opposite sides[J]. Applied Mathematics and Mechanics(English Edition), 2010, 31(10):1241-1250.
[6] Hou G L, Alatancang. Spectra of off-diagonal infinite dimensional Hamiltonian operators and their applications to plane elasticity problems[J]. Communications in Theoretical Physics, 2009, 51(2):200-204.
[7] Hou G L, Alatancang. Completeness of eigenfunction systems for off-diagonal infinite dimensional Hamiltonian operators[J]. Communications in Theoretical Physics, 2010, 53(2):237-241.
[8]侯国林,阿拉坦仓.无穷维Hamilton算子广义本征函数系的完备性及弹性力学中的应用[J].中国科学:数学,2012, 42(1):57-58.
[9] Qi G W, Hou G L, Alatancang. Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity[J]. Chinese Physics B, 2011, 20(12):264-272
[10] Wang H, Alatancang J J. Huang. Completeness of system of root vectors of upper triangular infinite-dimensional Hamiltonian operators appearing in elasticity theory[J]. Applied Mathematics and Mechanics(English Edition), 2012, 33(3):385-398.
[11]王元明.数学物理方程与特殊函数[M].北京:高等教育出版社,2013.
[12] Tolstov G P. Fourier series[M]. New York:Dover Publications, 1976
[13] Hou G L, Wa Q G, Xu Y N. Alatancang Chen.The separable Hamiltonian system and complete biorthogonal expansion method of Mindlin plate bending problems[J]. SCIENCE CHINA Physics,Mechanics&Astronomy, 2013, 56(5):974-980.
[2]姜礼尚,孙和生.偏微分方程选讲[M].北京:高等教育出版社,1997.
[3]钟万勰.弹性力学求解新体系[M].大连:大连理工大学出版社,1995.
[4]吴德玉,阿拉坦仓.无穷维Hamilton算子特征函数系的Cauchy主值意义下的完备性[J].中国科学:数学,2008, 38(8):904-912.
[5] Hou G L, Alatancang. Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supposed opposite sides[J]. Applied Mathematics and Mechanics(English Edition), 2010, 31(10):1241-1250.
[6] Hou G L, Alatancang. Spectra of off-diagonal infinite dimensional Hamiltonian operators and their applications to plane elasticity problems[J]. Communications in Theoretical Physics, 2009, 51(2):200-204.
[7] Hou G L, Alatancang. Completeness of eigenfunction systems for off-diagonal infinite dimensional Hamiltonian operators[J]. Communications in Theoretical Physics, 2010, 53(2):237-241.
[8]侯国林,阿拉坦仓.无穷维Hamilton算子广义本征函数系的完备性及弹性力学中的应用[J].中国科学:数学,2012, 42(1):57-58.
[9] Qi G W, Hou G L, Alatancang. Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity[J]. Chinese Physics B, 2011, 20(12):264-272
[10] Wang H, Alatancang J J. Huang. Completeness of system of root vectors of upper triangular infinite-dimensional Hamiltonian operators appearing in elasticity theory[J]. Applied Mathematics and Mechanics(English Edition), 2012, 33(3):385-398.
[11]王元明.数学物理方程与特殊函数[M].北京:高等教育出版社,2013.
[12] Tolstov G P. Fourier series[M]. New York:Dover Publications, 1976
[13] Hou G L, Wa Q G, Xu Y N. Alatancang Chen.The separable Hamiltonian system and complete biorthogonal expansion method of Mindlin plate bending problems[J]. SCIENCE CHINA Physics,Mechanics&Astronomy, 2013, 56(5):974-980.
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