Birkhoff系统的时间积分定理

研究Birkhoff系统的时间积分定理.建立Birkhoff系统的时间积分等式，由此等式导出系统的类功率方程，Pfaff-Birkhoff积分变分原理以及Pfaff-Birkhoff-d′Alembert微分变分原理. 关键词： Birkhoff系统 时间积分等式 类功率方程 变分原理     

广义Birkhoff方程的积分方法

场方法和最终乘子法是求解运动微分方程的基本方法.本文将这两种方法应用于广义Birkhoff系统,求出了场方法的基本偏微分方程和该方程的完全积分;根据Jacobi最终乘子定理求出了广义Birkhoff方程的解.并举例说明结果的应用.     

构造Birkhoff表示的广义Hojman方法

研究第一积分、Hojman方法及Birkhoff方程之间的内在联系. Hojman方法构造的Birkhoff函数(组)满足的一个特定关系, 对此关系加以分析得到更为一般的广义Hojman方法. 再将此关系与Birkhoff方程相结合, 导出Birkhoff系统Hojman意义下的循环积分. 举例说明结论的应用. 关键词： Birkhoff系统 Hojman方法 广义Hojman方法 循环积分     

广义Birkhoff系统的一类积分

将Birkhoff方程添加一附加项成为广义Birkhoff方程.将Birkhoff方程的一类积分推广并应用于广义Birkhoff方程.举例说明结果的应用. 关键词： Birkhoff方程 广义Birkhoff方程 积分     

一类广义Birkhoff系统的无限小正则变换与积分

研究一类广义Birkhoff系统的无限小正则变换与积分, 建立这类广义Birkhoff系统的方程,将Birkhoff系统的无限小正则变换与积分的有关结果推广到这类广义Birkhoff系统,举例说明结果的应用.     

转动相对论Birkhoff约束系统积分不变量的构造

研究转动相对论Birkhoff约束系统积分不变量的构造首先,建立转动相对论系统的约束Birkhoff方程;其次,利用等时变分与非等时变分之间的关系建立系统的非等时变分方程;然后,研究转动相对论Birkhoff约束系统的第一积分与积分不变量之间的关系,证明由系统的一个第一积分可以构造一个积分不变量,并给出自由Birkhoff系统的相应结果;最后,讨论转动相对论Hamilton系统、相对论Birkhoff系统和Hamilton系统、经典转动系统和等时变分情形下的积分不变量的构造,结果表明相关的结论均为该定理的特款给出一个例子说明结果的应用 关键词： 转动相对论 Birkhoff系统 约束 第一积分 积分不变量     

广义Birkhoff系统动力学的一类逆问题

针对广义Birkhoff系统动力学,提出广义Birkhoff系统动力学的一类逆问题,研究由已知积分流形来建立广义Birkhoff方程. 这类逆问题的解通常不是唯一的,需给出必要的补充要求. 最后举例说明结果的应用. 关键词： 广义Birkhoff系统 动力学逆问题 积分流形     

广义Birkhoff系统的积分不变量

研究广义Birkhoff系统的积分不变量.给出系统存在积分不变量的条件,在此条件下导出系统的线性积分不变量、通用积分不变量和二阶绝对积分不变量.举例说明结果的应用. 关键词： 广义Birkhoff方程 线性积分不变量 通用积分不变量 二阶绝对积分不变量     

转动系统相对论性Birkhoff动力学的基本理论

建立转动系统相对论性Birkhoff动力学的基本理论,给出其Birkhoff函数和Birkhoff函数组、Pfaff作用量、Pfaff-Birkhoff原理、Pfaff-Birkhoff-D’Alembert原理,以及Birkhoff方程.并研究转动系统相对论性Lagrange力学、Hamilton力学与转动系统相对论性Birkhoff动力学之间的关系,证明完整保守、完整非保守转动相对论系统都可纳入转动相对论Birkhoff系统 关键词： 转动系统 相对论 Birkhoff动力学 变分原理     

不同积分变分原理的统一

依据定量因果原理的数学表示，统一地导出了Lagrange量中含坐标关于时间一阶、二阶导数 的积分型的Hamilton原理、Voss原理、Hlder原理和Maupertuis-Lagrange原理等，给出了 这些原理的本质联系和统一描述.得出f₀=0并不是通常的保持Euler-Lagrange方 程不 变的结果，而是满足定量因果原理的结果.还得出Lagrange量的所有的积分型变分原理等价 地对应于两类满足定量因果原理的不变形式.同时发现所有积分型变分原理的运动方程都是E uler-Lagrange 方程，但不同条件的变分原理所对应的不同群Ｇ作用下的守恒量是不同 的.从而可对过去众多零散的积分型变分原理有一个系统和深入的理解，并使这些变分原理 自然地成为定量因果原理的推论. 关键词： 变分原理 因果原理 运动方程 对称性     

Dynamics of nonholonomic systems from variational principles embedded variation identity

Nondeterminacy of dynamics, i.e., the nonholonomic or the vakonomic, fundamental variational principles, e.g., the Lagrange-d'Alembert or Hamiltonian, and variational operators, etc., of nonholonomic mechanical systems can be attributed to the non-uniqueness of ways how to realize nonholonomic constraints. Making use of a variation identity of nonholonomic constraints embedded into the Hamilton's principle with the method of Lagrange undetermined multipliers, three kinds of dynamics for the nonholonomic systems including the vakonomic and nonholonomic ones and a new one are obtained if the variation is respectively reduced to three conditional variations: vakonomic variation, Hölder's variation and Suslov's variation, defined by the identity. Therefore, different dynamics of nonholonomic systems can be derived from an integral variational principle, utilizing one way of embedding constraints into the principle, with different variations. It is verified that the similar embedding of the identity into the Lagrange-d'Alembert principle gives rise to the nonholonomic dynamics but fails to give the vakonomic one unless the constraints are integrable.     

Conservation laws of nonconservative nonholonomic system based on Herglotz variational problem

The generalized variational principle of Herglotz type provides a variational method for describing nonconservative or dissipative processes. The purpose of this letter is to extend this variational principle to a first order linear nonholonomic system and study the conservation laws of the nonconservative nonholonomic system based on Herglotz variational problem. A new differential variational principle of the nonconservative nonholonomic system is proposed, which is based on Herglotz variational problem. And the differential equations of motion of the system are also obtained. Then, according to the condition for the invariance of the differential variational principle, the conservation theorem based on Herglotz variational problem for the nonconservative nonholonomic system are obtained. The theorem contains the conservation theorem of the nonconservative holonomic system as its special case, which can be reduced to the first Noether's theorem based on Herglotz variational problem under proper conditions. The inverse theorem of the conservation theorem is also provided and proved. An example is given to illustrate the application at the end of this letter.     

New block matrix spectral problem and Hamiltonian structure of the discrete integrable coupling system

In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity.     

Allgemeinrelativistisches Variationsprinzip für Flüssigkeiten und Festkörper in Eulerschen Koordinaten

The basic equations of the general relatiyistic continuum mechanics are derived in Eulerian coordinates by a variational principle with subsidiary conditions. For liquids and solids the Einsteinian field equations with the attached energy-momentum tensor and the adequate equations of motion for the mechanical and thermodynamical quantities follow. The subsidiary conditions are on the one hand physical requirements for system and processes (reversibility, particle conservation), on the other hand consequences of definitions (normalization of velocity). Lin's principle of the identity of particles is discussed. It is necessary as a subsidiary condition.     

A variational approach to stochastic nonlinear problems

A variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities. Approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a Bernoulli oscillator driven by white noise. Comparison with exact data available for this system shows that the variational approach to such problems can be quite effective.     

一类扰动洛伦兹系统的解法

研究了一个大气物理中洛伦兹系统的求解问题.首先利用广义变分原理构造一组变分迭代,其次决定系统的初始近似,最后通过变分迭代方法得到了对应模型的各次近似解.广义变分迭代方法是一个解析方法,得到的解还能够继续进行解析运算. 关键词： 洛伦兹方程 变分原理 近似解     

赤道东太平洋SST的海-气振子模型

研究了一个海-气振子的非线性耦合系统的模型.利用变分迭代原理，首先构造了相应的一组泛函.其次选取其Lagrange乘子.再用广义变分迭代方法得到了海-气振子模型一组解的近似序列. 关键词： 海-气振子 非线性 耦合系统 厄尔尼诺模型     

A new type of adiabatic invariants for disturbed Birkhoffian system of Herglotz type

The generalized variational principle of Herglotz type provides an effective way to study the problems of conservative and non-conservative systems in a unified way. According to the differential variational principle of Herglotz type, we study the adiabatic invariants for a disturbed Birkhoffian system in this paper. Firstly, the differential equations of motion of the Birkhoffian system based upon this variational principle are given, and the exact invariant of Herglotz type of the system is introduced. Secondly, a new type of adiabatic invariants for the system under the action of small perturbation is obtained. Thirdly, the inverse theorem of adiabatic invariant for the disturbed Birkhoffian system of Herglotz type is obtained. Finally, an example is given.     

Noether Symmetry and Conserved Quantities of Fractional Birkhoffian System in Terms of Herglotz Variational Problem

The aim of this paper is to study the Herglotz variational principle of the fractional Birkhoffian system and its Noether symmetry and conserved quantities. First, the fractional Pfaff-Herglotz action and the fractional PfaffHerglotz principle are presented. Second, based on different definitions of fractional derivatives, four kinds of fractional Birkhoff's equations in terms of the Herglotz variational principle are established. Further, the definition and criterion of Noether symmetry of the fractional Birkhoffian system in terms of the Herglotz variational problem are given. According to the relationship between the symmetry and the conserved quantities, the Noether's theorems within four different fractional derivatives are derived, which can reduce to the Noether's theorem of the Birkhoffian system in terms of the Herglotz variational principle under the classical conditions. As applications of the Noether's t heorems of the fractional Birkhoffian system in terms of the Herglotz variational principle, an example is given at the end of this paper.