关于Emden方程的对称性——————分析力学札记之十一

 研究著名的Emden方程的3种表达在群的无限小变换 下的Noether对称性、Lie对称性和形式不变性. 结果表明, 同一微分方程的不同表达可有不同的对称性.     

关于非完整系统的Lagrange型方程——————分析力学札记之十

 研究五类特殊非完整系统, 它们的运动微分方程有Lagrange形式.     

关于经典Appell-Hamel例——————分析力学札记之九

 不可积分微分约束系统动力学有Appell-Chetaev模型和Vacco模型,对经典Appell-Hamel例两个模型有同样解.     

关于理想约束和虚位移原理——分析力学札记之二十

 理想约束是分析力学的基本假定, 虚位移原理是分析静力学的基本原理. 本文从经典著作和重要教材中提取一些信息, 期望描述理想约束和虚位移原理的形成和发展, 并发表一些议论.     

关于Birkhoff方程和Lagrange方程--分析力学札记之十六

讨论Birkhoff方程和Lagrange方程的关系, 由Birkhoff方程可构造Lagrange方程, 反之, 由Lagrange方程可构造Birkhoff方程.     

关于动力学普遍方程——————理论力学札记之三

 归纳理论力学教材中关于动力学普遍方程的几种典型 表述,对这些表述进行评述, 并给出较全面的表述.     

关于机械能守恒律——————理论力学札记之一

 归纳理论力学教材中关于机械能守恒律的几种典型表述, 对典 型表述进行评述,并给出较全面表述的建议.     

关于牛顿力学的力和拉格朗日力学的力------理论力学札记之九

讨论了牛顿力学和拉格朗日力学中的力. 在牛顿力学中着重讨论了主动力, 约束和理想约束力. 在拉格朗日力学中着重讨论了广义力的两种分解方法.     

关于δq与δp的独立性问题——分析力学札记之廿二

 吴大猷先生指出,应用Hamilton原理时,坐标变分δq 系任意变化,而广义动量变分δp则非随意的. 有人认为δq与δp独立,还有人认为δq与δp可以看作是独立的. 整理一下各种表述,并提出管见.     

关于对动点的动量矩定理------理论力学札记之八

介绍质点系对动点的动量矩定理并讨论它的一些应用.     

非完整系统的Lie对称性与Noether对称性

研究非完整系统的Lie对称性与Noether对称性及其间的关系,具体研究了Chetaev型变量质量非完整系统和非Chetaev型非完整系统的Lie对称性与Noether对称性。给出Lie对称性导致Noether对称性及Noether对称性导致Lie对称性的条件。     

Periodic Solutions and Approximate Symmetries

Assuming that a dynamical system (DS) has a periodic solutionwith period T in the neighborhood of an equilibrium, a newDS can be obtained which governs approximately the motion oftrajectories in the neighborhood of the periodic orbit. Approximatesymmetries and first integrals have been discussed for this DS. Theresults have been demonstrated on the Helmholtz oscillator.     

Symmetries of Hamiltonian one-dimensional systems

The integrability of one-dimensional time-dependent particle Lagrangians is investigated via the techniques of Noether's theorem and the method of Lie symmetries.     

Elastic Symmetries of Defective Crystals

I construct discrete and continuous crystal structures that are compatible with a given choice of dislocation density tensor, and (following Mal’cev) provide a canonical form for these discrete structures. The symmetries of the discrete structures extend uniquely to symmetries of corresponding continuous structures—I calculate these symmetries explicitly for a particular choice of dislocation density tensor and deduce corresponding constraints on energy functions which model defective crystals.     

Rotational Symmetries of Crystals with Defects

I use the theory of Lie groups/algebras to discuss the symmetries of crystals with uniform distributions of defects.      

Approximate Transformation Groups and Renormgroup Symmetries

An approach that makes use of approximate group symmetries forcalculating renormgroup symmetries for boundary-value problems inmathematical physics is discussed. A comparison of exact and approximatesolutions for the nonlinear Shrödinger equation and quasi-Chaplyginequations is presented that demonstrates promising features of thesuggested approach.     

Application of Symmetries to Central Force Problems

We determine the Noether point symmetries associated with theusual Lagrangian of the differential equation of the orbit of the twobody problem. This gives rise to three definite natural forms (apartfrom the linear form) when the Lagrangian admits three Noether pointsymmetries which enables the solution of the orbit equation in terms ofelementary functions. The other forms for which the orbit equation hastwo point symmetries that arise in the Lie classification do not occurfor the usual Lagrangian. For central force problems we obtain inaddition to the six general central forces found by Broucke [8], newforce laws which lead to integrability in terms of known functions. Thisis achieved by the two Noether cases and further cases by use ofequivalence transformations of the orbit differential equation.Moreover, we give an extension of what is sometimes referred to asNewton's theorem of revolving orbits. We also make use of the Lieclassification to provide two new cases of integrable (in terms of knownfunctions) orbit differential equations and new force laws.     

Time-Reversible Hamiltonian Vector Fields with Symplectic Symmetries

This paper deals with the dynamics of time-reversible Hamiltonian vector fields with two degrees of freedom around an elliptic equilibrium point in presence of 1–1 resonance. The main result says that under certain conditions there are two one-parameter families of reversible periodic solutions terminating at the equilibrium.     

Symmetries and Hamiltonian Formalism for Complex Materials

Preliminary results toward the analysis of the Hamiltonian structure of multifield theories describing complex materials are reported: we invoke the invariance under the action of a general Lie group of the balance of substructural interactions. Poisson brackets are also introduced in the material representation to account for general material substructures. A Hamilton-Jacobi equation suitable for multifield models is presented. Finally, a spatial version of all these topics is discussed without making use of the notion of paragon setting.