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Under the frame of Hamiltonian, the propagation of seismic wave is a symplectic group transformation with one parameter and consequently, the numerical calculation methods of the propagation ought to be symplectic method.

在哈密顿表述下，地震波标量方程描述的波场随时间的演化过程为波场在单参数辛群作用下的变换过程。因而在计算地震波场随时间的演化时，为保持系统的辛性质，应采用辛几何算法。

Dual equations in Hamiltonian system are obttained and the original problem is reduced to the determination of eigenvalues and eigensolutions in the symplectic space.

以双板驱动空腔粘性流问题为研究对象，根据其特点建立了哈密顿体系下的对偶正则方程，将问题归结为辛体系下的本征值问题。

In this paper, the fundamental problems of the dynamic buckling in cylindrical shells are axial impact, torsion impact and coupling impact. The dual variables in symplectic space are gotten from energy function, and the Hamiltonian system is established.

本文针对弹性圆柱壳在轴向冲击、扭转冲击、轴扭耦合冲击及径向均布压力和轴向冲击载荷耦合作用下的动态屈曲问题，借助于系统的能量构造出问题的原变量和对偶变量，建立了系统的哈密顿体系。

When the Hamiltonian system is introduced into the research of dynamic buckling in cylindrical shells, the problem uses dual variables in Symplectic space instead of variables in traditional Euclidean space. The symplectic eigenfunction expansion method gives out an exact solution space. So many problems are solved, which can not find the solution in traditional methods.

将哈密顿体系引入结构的动态屈曲研究中，其意义在于将结构的动态屈曲研究从传统的欧几里得几何空间进入到由原变量和对偶变量组成的辛几何空间之中，从而使辛本征函数展开的直接解析法得以实施，这样就可以求解许多以往半逆凑合法无法求解的问题，并且所得到的解空间是完备的。

Viscoelastic method, which is described in the Euclidian space under the Lagrange system formulation, is updated to be in the Symplectic space under the conservation Hamiltonian system.

建立一套理论方法和数值辛算法。

A Hamiltonian system was constructed for the fundamental problem of the dynamic buckling ,thereby problems of critical loads and buckling modes were reduced to determinations of eigenvalues and eigensolutions in a symplectic space and a symplectic method was proposed.

针对有内压或外压的弹性圆柱壳在轴向冲击载荷耦合作用下的动态屈曲问题，构造哈密顿体系，在辛空间中将临界载荷和动态屈曲模态归结为辛本征值和本征解问题，从而形成一种辛方法。

Invariant Manifold: An unstable periodic orbit in a Hamiltonian system (a general class of differential equations which includes the three body problem) have associated two families of orbits forming tubes connected to the periodic orbit.

不变流形：一个不稳定的周期轨道的哈密顿系统（一个微分方程一般类，其中包括三个体问题）有关联的两个形成连接管轨道周期轨道的家庭。

For a system, Hamiltonian is invariant under any translation of spatial coordinates.

对于一体系，哈密算符在任何空间坐标变换下是不变的。

Unlike the LSD,the SIC-LSD Hamiltonian is not invariant undera unitary transformation among the occupied orbitals,which must be determined self-consistentlyfrom minimizing the SIC-LSD total energy.

与LSD近似不同，SIC-LSD近似的哈密顿量没有占据态的幺正变换不变性，单粒子态由SIC-LSD近似下的总能量极小决定。

Constructing a unitary matrix for transformation of coordinates by means of quadratic form theory,the Hamiltonian of the 3D coordinate-momentum coupling harmonic oscillator is transformed into diagonalmatrix,it not only offer a general mathematic method for solving this kind of problems,but also have an active effect upon upgrading students ability to solve physics problems by mathematical theory.

利用二次型理论构造一个幺正矩阵进行表象变换，将|x〉表象中的三模坐标-动量耦合量子谐振子体系的哈密顿量对角化，这不仅提供了一种解决该类问题的一般数学方法，同时对培养和提升学生运用数学工具解决复杂物理问题的能力也具有积极的指导作用。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。