Abstract:
As is well known, there are different Lagrangians which lead to the same Euler-Lagrange operator. The gauge invariance of a Lagrangian guarantees that of the corresponding Euler-Lagrange operator, but not vice versa. We show that the gauge invariance of an Euler-Lagrange operator, but not a Lagrangian results in Noether conservation laws.

Abstract:
By analogue with the methods and processes in continuous mechanics, a Lagrangian formulation and a Hamiltonian formulation of discrete mechanics are obtained. The dynamical equations including Euler--Lagrange equations and Hamilton's canonical equations of the discrete nonconservative holonomic systems are derived on a discrete variational principle. Some illustrative examples are also given.

Abstract:
We study the Euler-Lagrange cohomology and explore the symplectic or multisymplectic geometry and their preserving properties in classical mechanism and classical field theory in Lagrangian and Hamiltonian formalism in each case respectively. By virtue of the Euler-Lagrange cohomology that is nontrivial in the configuration space, we show that the symplectic or multisymplectic geometry and related preserving property can be established not only in the solution space but also in the function space if and only if the relevant closed Euler-Lagrange cohomological condition is satisfied in each case. We also apply the cohomological approach directly to Hamiltonian-like ODEs and Hamiltonian-like PDEs no matter whether there exist known Lagrangian and/or Hamiltonian associated with them.

Abstract:
The equations of Euler-Lagrange elasticity describe elastic deformations
without reference to stress or strain. These equations as previously published
are applicable only to quasi-static deformations. This paper extends these
equations to include time dependent deformations. To accomplish this, an
appropriate Lagrangian is defined and an extrema of the integral of this
Lagrangian over the original material volume and time is found. The result is a
set of Euler equations for the dynamics of elastic materials without stress or
strain, which are appropriate for both finite and infinitesimal deformations of
both isotropic and anisotropic materials. Finally, the resulting equations are
shown to be no more than Newton's Laws applied to each infinitesimal volume of
the material.

Abstract:
建立基于Lagrange-Euler方法的多液滴运动模型,数值模拟汽水分离器的分离效率和内部液滴湿度分布。根据多液滴运动特点,选取数目密度函数描述其运动行为。通过分析液滴和流场之间的作用机理,采用δ函数构建液滴运动参数等Lagrange变量和流场参数等Euler变量的联系,从而建立多液滴运动的物理模型,并给出该物理模型的数学描述及数值求解方法。利用该模型计算得到的分离器效率与实验值及商用软件模拟值符合良好,且通过对速度场的分析,认为计算所得的分离器内部湿度分布合理。结果表明: 多液滴运动模型所得结果可靠,该模型的提出为设计和优化分离器的结构提供了强有力的工具。 Abstract： The movement of polydispersed droplets was simulated using a Lagrangian-Eulerian model to calculate the separation efficiency and the moisture distributions in separators. A number density function was used to describe the motion. The mechanism describing the interaction between the droplets and the flow was developed with a δ-function that related the droplet and flow field parameters in a physical model of the polydispersed droplet motion. The mathematical description and numerical methods for solving the physical model was introduced. The efficiencies of the separators computed from this model fit well with experimental data and commercial CFD software model. The moisture distributions inside the separators predicted by the model are reasonable relative to the velocity fields. Thus, the results from the polydispersed droplet motion model are reliable and can be used to design and optimize separator designs.

Abstract:
建立基于Lagrange-Euler方法的多液滴运动模型,数值模拟汽水分离器的分离效率和内部液滴湿度分布。根据多液滴运动特点,选取数目密度函数描述其运动行为。通过分析液滴和流场之间的作用机理,采用δ函数构建液滴运动参数等Lagrange变量和流场参数等Euler变量的联系,从而建立多液滴运动的物理模型,并给出该物理模型的数学描述及数值求解方法。利用该模型计算得到的分离器效率与实验值及商用软件模拟值符合良好,且通过对速度场的分析,认为计算所得的分离器内部湿度分布合理。结果表明: 多液滴运动模型所得结果可靠,该模型的提出为设计和优化分离器的结构提供了强有力的工具。 Abstract： The movement of polydispersed droplets was simulated using a Lagrangian-Eulerian model to calculate the separation efficiency and the moisture distributions in separators. A number density function was used to describe the motion. The mechanism describing the interaction between the droplets and the flow was developed with a δ-function that related the droplet and flow field parameters in a physical model of the polydispersed droplet motion. The mathematical description and numerical methods for solving the physical model was introduced. The efficiencies of the separators computed from this model fit well with experimental data and commercial CFD software model. The moisture distributions inside the separators predicted by the model are reasonable relative to the velocity fields. Thus, the results from the polydispersed droplet motion model are reliable and can be used to design and optimize separator designs.

Abstract:
建立基于Lagrange-Euler方法的多液滴运动模型,数值模拟汽水分离器的分离效率和内部液滴湿度分布。根据多液滴运动特点,选取数目密度函数描述其运动行为。通过分析液滴和流场之间的作用机理,采用δ函数构建液滴运动参数等Lagrange变量和流场参数等Euler变量的联系,从而建立多液滴运动的物理模型,并给出该物理模型的数学描述及数值求解方法。利用该模型计算得到的分离器效率与实验值及商用软件模拟值符合良好,且通过对速度场的分析,认为计算所得的分离器内部湿度分布合理。结果表明: 多液滴运动模型所得结果可靠,该模型的提出为设计和优化分离器的结构提供了强有力的工具。 Abstract： The movement of polydispersed droplets was simulated using a Lagrangian-Eulerian model to calculate the separation efficiency and the moisture distributions in separators. A number density function was used to describe the motion. The mechanism describing the interaction between the droplets and the flow was developed with a δ-function that related the droplet and flow field parameters in a physical model of the polydispersed droplet motion. The mathematical description and numerical methods for solving the physical model was introduced. The efficiencies of the separators computed from this model fit well with experimental data and commercial CFD software model. The moisture distributions inside the separators predicted by the model are reasonable relative to the velocity fields. Thus, the results from the polydispersed droplet motion model are reliable and can be used to design and optimize separator designs.

Abstract:
建立基于Lagrange-Euler方法的多液滴运动模型,数值模拟汽水分离器的分离效率和内部液滴湿度分布。根据多液滴运动特点,选取数目密度函数描述其运动行为。通过分析液滴和流场之间的作用机理,采用δ函数构建液滴运动参数等Lagrange变量和流场参数等Euler变量的联系,从而建立多液滴运动的物理模型,并给出该物理模型的数学描述及数值求解方法。利用该模型计算得到的分离器效率与实验值及商用软件模拟值符合良好,且通过对速度场的分析,认为计算所得的分离器内部湿度分布合理。结果表明: 多液滴运动模型所得结果可靠,该模型的提出为设计和优化分离器的结构提供了强有力的工具。 Abstract： The movement of polydispersed droplets was simulated using a Lagrangian-Eulerian model to calculate the separation efficiency and the moisture distributions in separators. A number density function was used to describe the motion. The mechanism describing the interaction between the droplets and the flow was developed with a δ-function that related the droplet and flow field parameters in a physical model of the polydispersed droplet motion. The mathematical description and numerical methods for solving the physical model was introduced. The efficiencies of the separators computed from this model fit well with experimental data and commercial CFD software model. The moisture distributions inside the separators predicted by the model are reasonable relative to the velocity fields. Thus, the results from the polydispersed droplet motion model are reliable and can be used to design and optimize separator designs.

Abstract:
建立基于Lagrange-Euler方法的多液滴运动模型,数值模拟汽水分离器的分离效率和内部液滴湿度分布。根据多液滴运动特点,选取数目密度函数描述其运动行为。通过分析液滴和流场之间的作用机理,采用δ函数构建液滴运动参数等Lagrange变量和流场参数等Euler变量的联系,从而建立多液滴运动的物理模型,并给出该物理模型的数学描述及数值求解方法。利用该模型计算得到的分离器效率与实验值及商用软件模拟值符合良好,且通过对速度场的分析,认为计算所得的分离器内部湿度分布合理。结果表明: 多液滴运动模型所得结果可靠,该模型的提出为设计和优化分离器的结构提供了强有力的工具。 Abstract： The movement of polydispersed droplets was simulated using a Lagrangian-Eulerian model to calculate the separation efficiency and the moisture distributions in separators. A number density function was used to describe the motion. The mechanism describing the interaction between the droplets and the flow was developed with a δ-function that related the droplet and flow field parameters in a physical model of the polydispersed droplet motion. The mathematical description and numerical methods for solving the physical model was introduced. The efficiencies of the separators computed from this model fit well with experimental data and commercial CFD software model. The moisture distributions inside the separators predicted by the model are reasonable relative to the velocity fields. Thus, the results from the polydispersed droplet motion model are reliable and can be used to design and optimize separator designs.

Abstract:
We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense.