本文研究Birkhoff系统和广义Birkhoff系统平衡稳定性的动力学控制.首先建立系统的运动方程和平衡方程.其次,研究Birkhoff系统中控制参数出现在Birkhoff函数中平衡稳定性的动力学控制.方法是通过选取控制参数使得Birkhoff函数B成为定号函数,而其时间导数B为与B反号的常号函数.再次,研究广义Birkhoff系统平衡稳定性的动力学控制,通过选取Birkhoff函数或附加项中包含控制参数的方法,使得Birkhoff函数是定号函数,而其时间导数为反号的常号函数,从而控制系统的平衡稳定性.最后举例说明结果的应用.

理论力学中动力学普遍方程,在分析力学中称为d'Alembert-Lagrange原理.动力学普遍方程之普遍在于,由它不仅可导出动力学普遍定理,可导出完整约束系统和非完整约束系统的运动微分方程,还可导出积分变分原理.

德国女数学家Noether E 于1918年发表重要论文"不变变分问题".这篇论文给出两个定理,第一定理涉及经典力学的对称性与守恒量,第二定理涉及广义相对论.Noether第一定理不仅已成为研究经典力学和经典场论中,而且已成为研究量子力学和量子场论中对称性与守恒量关系的基础.本文介绍了Noether的这篇论文和她思想的传播,以及经典力学中的Noether定理.

1979年《力学与实践》创刊,至今已经走过 40年了.40 年来,《力学与实践》越办越好,已经成为广大力学工作者喜爱的刊物. 《力学与实践》为我们提供了发表力学教育与应用研究论文的园地.在非完整力学不被重视的年代,在朱照宣先生的支持下,在《力学与实践》上发表综述文章《非完整系统力学的历史与现状》(1979, 1(4): 6-10),为后来研究非完整力学打下了基础.

对于完整力学系统,若选取的参数不是完全独立的,则称为有多余坐标的完整系统.本文研究有多余坐标的可控力学系统的自由运动与初始运动.首先,需由d′Alembert-Lagrange原理并利用Lagrange乘子法建立有多余坐标完整系统的运动微分方程;其次,由约束系统自由运动的定义,令所有乘子为零,得到系统实现自由运动的条件.第三,如果给定运动的初始条件和控制参数,就可以研究系统的初始运动.文末,举例并说明方法和结果的应用.

Like the Hamilton-Jacobi method, the Vujanović field method transforms the problem of seeking the particular solution of an ordinary differential equations into the problem of finding the complete solution of a first order quasilinear partial differential equation, which is usually called the basic partial differential equation. Due to no need of the strong restrictive conditions required in the classic Hamilton-Jacobi method, the Vujanović field method may be used in many fields, such as non-conservative systems, nonholonomic systems, Birkhoff systems, controllable mechanical systems, etc. Even so, there is still a fundamental difficulty in the Vujanović field method. That is, for most of dynamical systems, it is hard to find the complete solution of the basic partial differential equation. In this paper, the Vujanović field method is improved into a new field method. The purpose of the improved field method is to find the first integrals of the motion equations, but not the particular solutions of the motion equations. The improved field method points out that for a basic partial differential equation with n independent variables, m (m n) first integrals of a dynamical system can be found as long as a solution with m arbitrary constants of the basic partial differential equation is found. In particular, if the complete solution (the complete solution is a special case of m=n) of the basic partial differential equation is found, all first integrals of the dynamical system can be found. That means that the motion of the dynamical system is completely determined. The Vujanović field method is just equivalent to this particular case. The improved field method expands the applicability of the field method, and is simpler than the Vujanović field method. Two examples are given to illustrate the effectiveness of the method. In addition, the improved field method is used to integrate the motion equations in Riemann-Cartan space. For a first-order linear homogenous scleronomous nonholonomic system subjected to an active force, its motion equation in its Riemann-Cartan configuration space can be obtained by a first order nonlinear nonholonomic mapping. Since the motion equations in Riemann-Cartan configuration space contain quasi-speeds, they are often considered to be difficult to solve directly. In this paper we give a briefing of how to construct the motion equations of a first order linear nonholonomic constraint system in its Riemann-Cartan configuration space, and how to obtain the first integrals of the motion equations in the Riemann-Cartan configuration space by the improved field method. This is an effective method to study some nonholonomic nonconservative motions.

外国人名不可乱译.汉译应遵循"约定成俗",或按所在国文字发音来译.有时可"拆开"来译,将一个音节拆成两个字.

理论力学的运动学部分有加速度合成定理:点的绝对加速度等于相对加速度、牵连加速度与科氏加速度的矢量和.在相对运动动力学部分有牵连惯性力和科氏惯性力.一般都说,科氏加速度和科氏力由科里奥利 (Coriolis GG, 1792—1843)于1835年首先提出.本文简述科氏力的历史与发展.

The geometric formulation of motion of the first-order linear homogenous scleronomous non-holonomic system subjected to active forces is studied with the nonholonomic mapping theory. The quasi-Newton law, the quasi-momentum theorem, and the second kind Lagrange equation of dynamical systems are obtained in the Riemann-Cartan configuration spaces. By the nonholonomic mapping, a Euclidean configuration space or a Riemann configuration space of a dynamical system can be mapped into a Riemann-Cartan configuration space with torsion. The differential equations of motion of the dynamical system can be obtained in its Riemann-Cartan configuration space by the quasi-Newton law or the quasi-momentum theorem. For a constrained system, the differential equations of motion in its Riemann-Cartan configuration space may be simpler than the equations in its Euclidean configuration space or its Riemann configuration space. Therefore, the nonholonomic mapping theory can solve some constrained problems, which are difficult to be solved by the traditional analytical mechanics method. Three examples are given to illustrate the effectiveness of the method.

With the development of science and technology,it is more and more important to study the dynamics of variable mass system such as jet aircraft and rocket,and it is always hoped that the solutions of the variable mass system are stable or asymptotically stable.It is difficult to study the stability by using Lyapunov direct methods because of the difficulty of constructing Lyapunov functions directly from the differential equations of the mechanical system.This paper presents an indirect method for studying stability,that is,gradient system method.This method can not only reveal the internal structure of dynamic system,but also help to explore the dynamic behavior such as the stability,asymptotic and bifurcation.The functionV of the gradient system is usually taken as a Lyapunov function,so the gradient system is more suitable to be studied with the Lyapunov function.The equations of motion for the holonomic mechanical system with variable mass are listed,and all generalized accelerations are obtained in the case of non-singular system.A class of gradient system with negative-definite matrix is proposed,and the stability of the solutions of the gradient system is studied.This kind of gradient system and variable mass mechanical system are combined,then the conditions under which the solutions of the mechanical systems with variable mass can be stable or asymptotically stable are given.Further the mechanical system with variable mass whose solution is stable or asymptotically stable is constructed by using the gradient system with non-symmetrical negative-definite matrix.Through specific examples,it is studied that the solutions of the single degree of freedom motion of a variable mass system are stable or asymptotically stable under some conditions of the laws of mass change,particle separation velocity and force.The method is also suitable for the study of other constrained mechanical systems.

提出两类广义梯度系统,即广义斜梯度系统和具有对称负定矩阵的广义梯度系统,并研究了这两类广义梯度系统的性质.在满足一定的条件下,把Tzénoff方程化成这两类广义梯度系统方程.进一步利用这两类广义梯度系统的性质来研究Tzénoff方程解的稳定性,分别给出解是稳定、渐进稳定和不稳定的条件,并举例说明结果的应用.

This paper first formulates a Hamiltonian system with hyperchaotic phenomena and investigates the equilibrium point and double Hopf bifurcation of the system. We obtain the result that the Hamiltonian system has hyperchaotic behaviors when any system parameter varies. The influences of holonomic constraint and nonholonomic constraint on the equilibrium points, invariance and the hyperchaotic state of the Hamiltonian system are then studied. Finally, we achieve the hyperchaotic control of the Hamiltonian system by introducing the constraint method. The studies indicate that the constraint can not only change the Hamiltonian system from hyperchaotic state to periodic state or chaotic state, but also make the Hamiltonian system become globally asymptotically stable. Numerical simulations, including Lyapunov exponents, bifurcation diagrams, Poincaré maps and phase portraits for systems, exhibit the complex dynamical behaviors.

The Hamilton-Jacobi equation is an important nonlinear partial differential equation. In particular, the classical Hamilton-Jacobi method is generally considered to be an important means to solve the holonomic conservative dynamics problems in classical dynamics. According to the classical Hamilton-Jacobi theory, the classical Hamilton-Jacobi equation corresponds to the canonical Hamilton equations of the holonomic conservative dynamics system. If the complete solution of the classical Hamilton-Jacobi equation can be found, the solution of the canonical Hamilton equations can be found by the algebraic method. From the point of geometry view, the essential of the Hamilton-Jacobi method is that the Hamilton-Jacobi equation promotes the vector field on the cotangent bundle T* M to a constraint submanifold of the manifold T* M R, and if the integral curve of the promoted vector field can be found, the projection of the integral curve in the cotangent bundle T* M is the solution of the Hamilton equations. According to the geometric theory of the first order partial differential equations, the Hamilton-Jacobi method may be regarded as the study of the characteristic curves which generate the integral manifolds of the Hamilton 2-form . This means that there is a duality relationship between the Hamilton-Jacobi equation and the canonical Hamilton equations. So if an action field, defined on UI (U is an open set of the configuration manifold M, IR), is a solution of the Hamilton-Jacobi equation, then there will exist a differentiable map from MR to T* MR which defines an integral submanifold for the Hamilton 2-form . Conversely, if * =0 and H1(UI)=0 (H1(UI) is the first de Rham group of U I), there will exist an action field S satisfying the Hamilton-Jacobi equation. Obviously, the above mentioned geometric theory can not only be applicable to the classical Hamilton-Jacobi equation, but also to the general Hamilton-Jacobi equation, in which some first order partial differential equations correspond to the non-conservative Hamiltonian systems. The geometry theory of the Hamilton-Jacobi method is applied to some special non-conservative Hamiltonian systems, and a new Hamilton-Jacobi method is established. The Hamilton canonical equations of the non-conservative Hamiltonian systems which are applied with non-conservative force Fi = (t)pi can be solved with the new method. If a complete solution of the corresponding Hamilton-Jacobi equation can be found, all the first integrals of the non-conservative Hamiltonian system will be found. The classical Hamilton-Jacobi method is a special case of the new Hamilton-Jacobi method. Some examples are constructed to illustrate the proposed method.

In this paper, we generalize the Pfaff–Birkhoff principle to the case of containing fractional derivatives and obtain the so-called fractional Pfaff–Birkhoff–d’Alembert principle. The fractional Birkhoff equations in the sense of Riemann–Liouville fractional derivative are derived. Under the framework of variational integrators, we develop the discrete fractional Birkhoff equations by approximating the Riemann–Liouville fractional derivative with the shifted Grünwald–Letnikov fractional derivative. The resulting algebraic equations can be served as an algorithm to numerically solve the fractional Birkhoff equations. A numerical example is demonstrated to show the validity and applicability of the presented methodology.

In order to study the integration and the stability of autonomous Birkhoffian systems, we propose four kinds of gradient systems to represent the autonomous Birkhoffian systems. By analysing the relationship between the gradient systems and the Birkhoffian systems, we obtain the conditions that the Birkhoffian systems can be transformed into a kind of four gradient systems. Then, we use the properties of gradient system to investigate the problems of integration and stability of the Birkhoffian systems. Finally, we give some examples to illustrate the application of the theory.

This paper first formulates a new hyperchaotic system for particle motion and analyzes the equilibrium stability of the system and the hyperchaotic behaviors in the motion of the particle on a horizontal smooth plane. We then investigate the influences of two nonlinear nonholonomic constraints on the particle motion. The numerical simulations, including bifurcation diagrams, computation of Lyapunov exponents, Poincaré maps, and phase portraits for systems, not only show the hyperchaotic phenomena, but also exhibit the different hyperchaotic behaviors in the constrained parameters space for the various regimes. Additionally, a holonomic constraint is introduced to control the hyperchaotic system.

Whittaker first put forward a new approach, called the initial motions, to solve the differential equations of motion aimed at holonomic systems. Since most of the differential equations of motion for mechanical systems are nonlinear ordinary ones, which are difficult to find the analytic solutions. Fortunately, the concept of initial motions can manage these situations and study its subsequent motions. This work is devoted to discuss and investigate the initial motions for mechanical systems, particularly for nonholonomic systems. First, the differential equations for holonomic systems are formulated, and the formulation and solution of initial motions of the systems are proposed. Second, the differential equations of motion for nonholonomic systems are established, based on the new method of initial motions to obtain the initial values of high-order derivatives of generalized velocities, the formulation and solution of initial motions are introduced in the general nonholonomic systems and Chaplygin systems. The methods and results obtained are illustrated by a number of classical examples, both for holonomic and nonholonomic systems.

Dependance of stability of equilibrium of generalized Birkhoff system on two parameters is studied.The differential equations of the generalized Birkhoff system and the gradient system are given respectively,and the conditions for the transformation of the generalized Birkhoff system into a gradient system are obtained.Under the conditions the generalized Birkhoff system is considered as a gradient system,and the characteristic of the gradient system can be used to study the stability and its dependance on two parameters of the system.The results show that the equilibrium stability of the system is likely to be stable,or asymptotically stable,and may be unstable with the change of two parameters.The stability region and unstable region are drawn on the parameter plane.Some examples are given to illustrate the application of the results.

When the mechanical system is coupled with the constraints,it is very convenient for the transition from the rectangular coordinates to the generalized coordinates,and it is also very necessary.The introduction of generalized coordinates is one of the major features of analytical mechanics,and the Lagrange equation is based on the generalized coordinates.In this paper,some relevant history data of the formation of generalized coordinates are provided,and some propositions are given.

The Ценов equations and Mac Millan equations are motion equations later appearing in the history of nonholonomic mechanics.These equations not only have the historical significance,but also have the advantage of theory and application.Some relevant history data are provided for the formation and the development of the Ценов equations and Mac Millan equations,and our propositions are given.