A new method based on the harmonic balance method for nonlinear oscillators

The harmonic balance (HB) method as an analytical approach is widely used for nonlinear oscillators, in which the initial conditions are generally simplified by setting velocity or displacement to be zero. Based on HB, we establish a new theory to address nonlinear conservative systems with arbitrary initial conditions, and deduce a set of over-determined algebraic equations. Since these deduced algebraic equations are not solved directly, a minimization problem is constructed instead and an iterative algorithm is employed to seek the minimization point. Taking Duffing and Duffing-harmonic equations as numerical examples, we find that these attained solutions are not only with high degree of accuracy, but also uniformly valid in the whole solution domain.     

非线性非完整系统Raitzin正则方程的Hojman守恒定理

利用时间不变的无限小变换下的Lie对称性，研究非线性非完整系统Raitzin正则方程的Hojman守恒定理.列出系统的运动微分方程.建立时间不变的无限小变换下的确定方程.给出系统的Hojman守恒定理，并举例说明结果的应用. 关键词： 非线性非完整系统 Raitzin正则方程 Lie对称性 确定方程 Hojman守恒 定理     

Structural Equation and Mei Conserved Quantity of Mei Symmetry forAppell Equations in Holonomic Systems with Unilateral Constraints

Structural equation and Mei conserved quantity of Mei symmetry for Appell equations in holonomic systems with unilateral constraints are investigated. Appell equations and differential equations of motion for holonomic mechanic systems with unilateral constraints are established. The definition and the criterion of Mei symmetry for Appell equations in holonomic systems with unilateral constraints under the infinitesimal transformations of groups are also given. The expressions of the structural equation and Mei conserved quantity of Mei symmetry for Appell equations in holonomic systems with unilateral constraints expressed by Appell functions are obtained. An example is given to illustrate the application of the results.     

Generalized Conformal Symmetries and Its Application of Hamilton Systems

In this paper the generalized conformal symmetries and conserved quantities by Lie point transformations of Hamilton systems are studied. The necessary and sufficient conditions of conformal symmetry by the action of infinitesimal Lie point transformations which are simultaneous Lie symmetry are given. This kind type determining equations of conformal symmetry of mechanical systems are studied. The Hojman conserved quantities of the Hamilton systems under infinitesimal special transformations are obtained. The relations between conformal symmetries and the Lie symmetries are derived for Hamilton systems. Finally, as application of the conformal symmetries, an illustration example is introduced.     

变质量完整力学系统的非Noether守恒量

利用时间不变的无限小变换下的Lie对称性,研究变质量完整力学系统的一类新的守恒量.给出系统的运动微分方程,研究时间不变的无限小变换下的Lie对称性确定方程,将Hojman定理推广并应用于这类系统 关键词： 变质量系统 完整约束 确定方程 非Noether守恒量     

A type of new conserved quantity deduced from Mei symmetry for Nielsen equations in a holonomic system with unilateral constraints

<正>A type of new conserved quantity deduced from Mei symmetry for Nielsen equations in a holonomic system with unilateral constraints is investigated.Nielsen equations and differential equations of motion for the holonomic mechanical system with unilateral constraints are established.The definition and the criterion of Mei symmetry for Nielsen equations in the holonomic systems with unilateral constraints under the infinitesimal transformations of Lie group are also given.The expressions of the structural equation and a type of new conserved quantity of Mei symmetry for Nielsen equations in the holonomic system with unilateral constraints are obtained.An example is given to illustrate the application of the results.     

Nonlocal conserved quantities,balance laws,and equations of motion

A formalism is developed whereby balance laws are directly obtained from nonlocal (integrodifferential) linear second-order equations of motion for systems described by several dependent variables. These laws augment the equations of motion as further useful information about the physical system and, under certain conditions, are shown to reduce to conservation laws. The formalism can be applied to physical systems whose equations of motion may be relativistic and either classical or quantum. It is shown to facilitate obtaining global conservation laws for quantities which include energy and momentum. Applications of the formalism are given for a nonlocal Schrödinger equation and for a system of local relativistic equations of motion describing particles of arbitrary integral spin.     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

Molecular,metabolic, and genetic control: An introduction

The living cell is a miniature, self-reproducing, biochemical machine. Like all machines, it has a power supply, a set of working components that carry out its necessary tasks, and control systems that ensure the proper coordination of these tasks. In this Special Issue, we focus on the molecular regulatory systems that control cell metabolism, gene expression, environmental responses, development, and reproduction. As for the control systems in human-engineered machines, these regulatory networks can be described by nonlinear dynamical equations, for example, ordinary differential equations, reaction-diffusion equations, stochastic differential equations, or cellular automata. The articles collected here illustrate (i) a range of theoretical problems presented by modern concepts of cellular regulation, (ii) some strategies for converting molecular mechanisms into dynamical systems, (iii) some useful mathematical tools for analyzing and simulating these systems, and (iv) the sort of results that derive from serious interplay between theory and experiment. (c) 2001 American Institute of Physics.     

Quantum Maps with Memory from Generalized Lindblad Equation

In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.     

Role of local environment relaxation in calculating the reaction rates for nonideal reaction systems

An analysis of the equations for the rates of elementary stages in nonideal reaction systems demonstrates that the traditional notion of existence of the chemical potential of the activated complex in dense phases is inconsistent with the actual times of molecular motions and with the procedure of statistical averaging of the molecular distributions. Expressions relating the rate constants of elementary chemical reactions for experiments at atmospheric pressure and under supercritical conditions are derived.     

Lagrange equations of nonholonomic systems with fractional derivatives

This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton's principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d'Alembert–Lagrange principle with fractional derivatives is presented,and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.     

Construction of the solution of variational equations for constrained Birkhoffian systems

In this paper we present the variational equations of constrained Birkhoffian systems and study their solution.It is proven that,under some conditions,a particular solution of variational equations can be obtained by using a first integral.At the end of the paper,an example is given to illustrate the application of the results.     

准坐标下广义非保守系统Lagrange方程的积分因子与守恒定理

用积分因子方法研究准坐标下广义非保守系统Lagrange方程的守恒定理.列写系统的运动微分方程，给出它的积分因子的定义.研究守恒量存在的必要条件.建立系统的守恒定理及其逆定理，并举例说明结果的应用. 关键词： 准坐标 Lagrange方程 积分因子 守恒量     

事件空间中Birkhoff系统的参数方程及其第一积分

研究事件空间中Birkhoff系统动力学.在(2n+1)维事件空间中,建立了Birkhoff系统的Pfaff-Birkhoff-d'Alembert原理和Birkhoff参数方程,研究了方程的第一积分,给出了第一积分及其存在条件. 关键词： Birkhoff系统 事件空间 参数方程 第一积分     

一类非线性色散Boussinesq方程的隐式孤立波解

用动力系统分岔方法研究了一类非线性色散Boussinesq方程.在不同的参数条件下,给出了该方程具有隐函数形式的孤立波解的解析表达式.数值模拟进一步验证了所得结果的正确性.     

The role of mathematics in gravitational physics

Certain gravitational systems are described for which particular equations of state are singled out by means of purely mathematical considerations. The fact that these equations of state are realistic, and indeed the most relevant to the physical systems under study, suggests that locked into Einstein's field equations lies some information about local physical conditions, and that in order to describe gravitating systems, it is not always necessary to draw on extraneous branches of physics.This essay received an honorable mention (1976) from the Gravity Research Foundation-Ed.     

Nonthermodynamic fluctuations in two-dimensional hydrodynamic systems

The problem of fluctuations in two-dimensional hydrodynamic systems under nonequilibrium conditions due to external forces is studied on the example of a simple model. The effective action is given; this action makes it possible to construct a perturbation series to calculate fluctuation effects in such systems. Renormalizability of this action with an infinite number of vertices is proved; renormalization group equations are derived. A number of solutions of these equations are found; the asymptotic behaviour of the solution is considered in the limit of large logarithms.     

Structure equation and Mei conserved quantity for Mei symmetry of Appell equation

This paper investigates structure equation and Mei conserved quantity of Mei symmetry of Appell equations for non-Chetaev nonholonomic systems. Appell equations and differential equations of motion for non-Chetaev nonholonomic mechanical systems are established. A new expression of the total derivative of the function with respect to time t along the trajectory of a curve of the system is obtained, the definition and the criterion of Mei symmetry of Appell equations under the infinitesimal transformations of groups are also given. The expressions of the structure equation and the Mei conserved quantity of Mei symmetry in the Appell function are obtained. An example is given to illustrate the application of the results.     

Structure equation and Mei conserved quantity for Mei symmetry of Appell equation

This paper investigates structure equation and Mei conserved quantity of Mei symmetry of Appell equations for non-Chetaev nonholonomic systems. Appell equations and differential equations of motion for non-Chetaev nonholonomic mechanical systems are established. A new expression of the total derivative of the function with respect to time $t$ along the trajectory of a curve of the system is obtained, the definition and the criterion of Mei symmetry of Appell equations under the infinitesimal transformations of groups are also given. The expressions of the structure equation and the Mei conserved quantity of Mei symmetry in the Appell function are obtained. An example is given to illustrate the application of the results.