非Четаев型非完整系统的Lie对称性与守恒量

研究非Четаев型非完整系统的Ｌｉｅ对称性．首先利用微分方程在无限小变换下的不变性建立Ｌｉｅ对称所满足的确定方程和限制方程，给出结构方程并求出守恒量；其次研究上述问题的逆问题：根据已知积分求相应的Ｌｉｅ对称性；最后举例说明结果的应用．     

Application of Lie transformation group methods to classical linear theories of rods and plates

In the present paper, a class of partial differential equations governing various rod and plate theories of Bernoulli–Euler and Poisson–Kirchhoff type is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived and the general statement of the associated group-classification problem is given. A simple relation is deduced allowing to recognize easily the variational symmetries among the “ordinary” symmetries of a self-adjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Bessel-Hagen’s extension of Noether’s theorem) conservation laws are suggested. Solutions of group-classification problems are given for subclasses of equations of the foregoing type governing stability and vibration of rods, fluid conveying pipes and plates resting on variable elastic foundations. The obtained group-classification results are used to derive conservation laws and group-invariant solutions readily applicable in rod dynamics and plate statics and dynamics. New generalized symmetries and conservation laws for the theories of Timoshenko beams, Reissner–Mindlin plates and three-dimensional elastostatics are presented.     

时间尺度上Lagrange系统的Hojman守恒量

利用对称性和守恒律, 可以简化动力学问题甚至求解力学系统的精确解, 更好地理解其动力学行为. 时间尺度分析将连续和离散动力学模型统一并拓展到时间尺度框架, 既避免了重复研究又可揭示两者之区别和联系. 因此, 通过对称性来探寻在时间尺度的框架下新的守恒定律很有必要. 本文首先建立了时间尺度上Lagrange方程, 利用时间尺度微积分性质导出了时间尺度上Lagrange系统的两个重要关系式; 其次, 依据微分方程在单参数Lie变换群下的不变性, 建立了时间尺度上Lie对称性的定义和确定方程; 最后, 建立了时间尺度上Lie对称性定理并利用上述关系式给出了证明, 得到了时间尺度上Lagrange系统的新守恒量. 当时间尺度取为实数集时, 该守恒量退化为著名的Hojman守恒量. 文末考察了一个两自由度时间尺度Lagrange系统, 在3种不同时间尺度情形下得到了该系统的Hojman守恒量, 数值计算结果验证了定理的正确性.      

A unified theory for turbulent wake flows described by eddy viscosity

In this paper, we consider the conservation laws for the far downstream wake equations described by eddy viscosity. A basis of conserved vectors is constructed. The well-known conserved quantities for the turbulent classical wake and the turbulent wake of a self-propelled body are obtained by integrating the corresponding conservation law across the wake and imposing the boundary conditions. For the wake of a self-propelled body the additional condition that the drag on the body is zero and is required to obtain the conserved quantity. A third conservation law, which possibly belongs to another type of wake, is discovered. The Lie point symmetry associated with the conserved vector is used to obtain the invariant solution and a typical velocity profile for this wake is provided. This wake appears to have common properties with the other two well-known wakes. We then analyse the invariant solutions to all three wake problems and prove that a simple mathematical relationship exists between them thus unifying the theory for turbulent wake flows.     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

Solitons and conservation laws of Klein–Gordon equation with power law and log law nonlinearities

This paper obtains the conservation laws of the Klein–Gordon equation with power law and log law nonlinearities. The multiplier approach with Lie symmetry analysis is employed to obtain the conserved densities. The 1-soliton solutions are subsequently used to compute the conserved quantities from the conserved densities. Later the perturbation terms are added and the conservation laws of the perturbed Klein–Gordon equation are studied.     

A new Lie symmetrical method of finding conserved quantity for Birkhoffian systems

For a Birkhoffian system, a new Lie symmetrical method to find a conserved quantity is given. Based on the invariance of the equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, several important relationships which reveal the interior properties of the Birkhoffian system are given. By using these relationships, a new Lie symmetrical conservation law for the Birkhoffian system is presented. The new conserved quantity is constructed in terms of infinitesimal generators of the Lie symmetry and the system itself without solving the structural equation which may be very difficult to solve. Furthermore, several deductions are given in the special infinitesimal transformations and the results are reduced to a Hamiltonian system. Finally, one example is given to illustrate the method and results of the application.     

Hamilton系统的一类新型守恒律

研究Hamilton系统的Lie对称性与守恒律。根据微分方程在无限小群变换下的不变性理论，建立了Hamilton系统仅依赖于正则变量的无限小群变换的Lie对称变换，给出了Lie对称性的确定方程，并直接由系统的Lie对称性得到了系统的一类新型定恒律。文末，举例说明结果的应用。     

On group properties and conservation laws for second-order quasi-linear differential equations

A sufficient condition for the absence of tangent transformations admitted by second-order quasi-linear differential equations and a sufficient condition for linear autonomy of operators of the Lie group of transformations admitted by second-order weakly nonlinear differential equations are found. A theorem on the structure of the first-order conservation laws for second-order weakly nonlinear differential equations is proved. A classification of second-order linear differential equations with two independent variables in terms of first-order conservation laws is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 64–70, May–June, 2009.     

New <Emphasis Type="Italic">higher-order</Emphasis> conservation laws of some classes of wave and Gordon-type equations

A large class of wave equations, with dissipation and source terms (Gordon type equations), are analysed using a symmetry approach and constructing conservation laws. We obtain some, previously unknown, relationships between the conservation laws and symmetries in the former case. In the latter case, we use the multiplier (and homotopy) approach to construct conservation laws from which some surprisingly, interesting higher-order variational symmetries and corresponding conserved quantities are obtained for a large class of Gordon type equations similar to those of the sine-Gordon equation.     

Potential symmetries and conservation laws for generalized quasilinear hyperbolic equations

Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physically interesting cases are found by using the partial Lagrangian approach.     

Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians

We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived. Examples are given including those that admit a standard Lagrangian such as the Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations in more than two independent variables.     

Lie symmetry and approximate Hojman conserved quantity of Appell equations for a weakly nonholonomic system

For a weakly nonholonomic system, the Lie symmetry and approximate Hojman conserved quantity of Appell equations are studied. Based on the Appell equations for a weakly nonholonomic system under special infinitesimal transformations of a group in which the time is invariable, the definition of the Lie symmetry of the weakly nonholonomic system and its first-degree approximate holonomic system are given. With the aid of the structure equation that the gauge function satisfies, the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are derived. Finally, an example is given to study the exact and approximate Hojman conserved quantity of the system.     

变质量非完整系统的Lie对称性逆问题

本文研究质量非完整系统的Lie对称性逆问题：根据已知积分求相应的Lie对称性,具体研究了受Chetaev型和非Chetaev型非完整约束的变质量系统的Lie对称性逆问题。首先,根据Lie对称所满足的确定方程和限制方程,给出Lie对称的结构方程和相应的守恒量及其表达式;其次,由已知守恒量求出相应的Noether对称性;最后,根据Noether对称性求出相应的Lie对称性。     

变长度弹性伸缩腿双足机器人动力学与控制

研究了半被动双足机器人的平面稳定行走的控制问题．基于弹簧质点模型，采用拉格朗日方法分别得到双足机器人单支撑阶段与双支撑阶段的动力学方程，对机器人系统的动力学方程求得周期解．应用非线性系统状态反馈线性化理论，在双足机器人的单支撑阶段和双支撑阶段中，通过控制双足机器人的腿长度，实现稳定的周期行走．在理论分析的基础上，对控制算法进行了仿真与研究．结果表明：在周期行走过程中，文中采用的变长度控制算法可以使双足机器人克服外界的干扰，并具有较强的抗干扰性．     

Input torque sensitivity to uncertain parameters in biped robot

Input torque is the main power to maintain bipedal walking of robot, and can be calculated from trajectory planning and dynamic modeling on biped robot. During bipedal walking, the input torque is usually required to be adjusted due to some uncertain parameters arising from objective or subjective factors in the dynamical model to maintain the pre-planned stable trajectory. Here, a planar 5-link biped robot is used as an illustrating example to investigate the effects of uncertain parameters on the input torques. Kine-matic equations of the biped robot are firstly established by the third-order spline curves based on the trajectory planning method, and the dynamic modeling is accomplished by taking both the certain and uncertain parameters into account. Next, several evaluation indices on input torques are intro-duced to perform sensitivity analysis of the input torque with respect to the uncertain parameters. Finally, based on the Monte Carlo simulation, the values of evaluation indices on input torques are presented, from which all the robot param-eters are classified into three categories, i.e., strongly sensi-tive, sensitive and almost insensitive parameters.     

Optimal system,group invariant solutions and conservation laws of the CGKP equation

In this paper, the (2 + 1)-dimensional cubic generalized Kadomtsev–Petviashvili (CGKP) equation that is derived from the Maxwell–Bloch equations is investigated. By means of Lie symmetry analysis method, we obtain the Lie point symmetries for the equation and the optimal system of the symmetry algebra. Based on the optimal system, a lot of group invariant solutions are obtained. In addition, explicit conservation laws of the equation are studied.     

On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.     

Exact solutions of thin film flows

This article studies the analytical solutions for two thin film flow problems on a moving belt. The reduction of the equations follows from their Lie point symmetry generators and conservation laws which are valid for the considered boundary conditions also. The solutions for the two problems are developed using the correct and nonlinear boundary condition for the free surface. Mathematica is adopted for some of the analysis.     

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.