Exact invariants and adiabatic invariants of Raitzin＇s canonical equations of motion for a nonlinear nonholonomic mechanical system

The exact invariants and the adiabatic invariants of Raitzin＇s canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher-order adiabatic invariant of a mechanical system under the action of a small perturbation, the forms of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results.

Variation principle of piezothermoelastic bodies,canonical equation and homogeneous equation

Combining the symplectic variations theory, the homogeneous control equation and isopaxametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced. Firstly, based on the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered, and the non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation. For the convenience of deriving Hamilton isopaxametric element formulations with four nodes, one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle. The homogeneous equation simplifies greatly the solution programs which axe often performed to solve nonhomogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.

Three-order pseudo-Hamilton canonical equationsReceived

Based on the three-order Lagrangian equations, Hamilton＇s function of acceleration H^＊ and generalized acceleration momentum P^＊α are defined, and pseudo-Hamilton canonical equations corresponding to three-order Lagrangian equations are obtained. The equations are similar to Hamilton＇s canonical equations of analytical mechanics in form.

Form Invariance of Raitzin＇s Canonical Equations of Relativistic Mechanical System

A form invariance of Raitzin's canonical equations of relativistic mechanical system is studied. First, the Raitzin's canonical equations of the system are established. Next, the definition and criterion of the form invariance in the system under infinitesimal transformations of groups are given. Finally, the relation between the form invariance and the conserved quantity of the system is obtained and an example is given to illustrate the application of the result.

Exact Invariants and Adiabatic Invariants of Raitzin＇s Canonical Equations of Motion for Nonholonomic System of Non-Chetaev＇s Type

The exact invariants and the adiabatic invariants of Raitzin's canonical equations of motion for the nonholonomic system of non-Chetaev's type are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher order adiabatic invariant of mechanical system with the action of a small perturbation, the form of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results.

An algorithm and its application for obtaining some kind of infinite-dimensional Hamiltonian canonical formulation

Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical ＇σ/σx＇-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.

Fractional differential equations of motion in terms of combined Riemann-Liouville derivatives

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

The Explicit Solution to Equation AX ＋ X B = C in Matrices

At first one of g-inverses of A(?) I_n+I_m(?) B^Tis given out,then the explicit solution to matrix equation AX+XB=C is gained by using the method of matrix decomposition, finally,a numerical example is obtained.

On existence and uniqueness of solutions to uncertain backward stochastic differential equations

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.

The Reflexive Solutions of the Matrix Equations (AX,XB^H )= (C,D^H )

The matrix equations (AX, XBH)=(C, DH) have been widely used in structural design, parameter identification, linear optimal control, and so on. But few researches studied the reflexive solutions. A new approach for the reflexive solutions to the matrix equations was proposed. By applying the canonical correlation decomposition (CCD) of matrix pairs, the necessary and sufficient conditions for the existence and the general expression for the reflexive solutions of the matrix equations (AX, XBH)=(C, DH) were established. In addition, by using the methods of space decomposition, the expression of the optimal approximation solution to a given matrix was derived.

Hamiltonian Formulation for Water Wave Equation

This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressible, inviscid fluid have been constructed in cartesian coordinates and also in cylindrical coordinates. Then Lagrangian function within a certain flow region is expanded under the assumption that the dispersion μ and the nonlinearity ε satisfied . Using Hamilton’s principle for water wave evolution Hamiltonian formulation is derived. It is obvious that the motion of the system is conservative. Then Hamilton’s canonical equation of motion is also derived.

Exact Quasi-Classical Asymptotic beyond Maslov Canonical Operator and Quantum Jumps Nature

Exact quasi-classical asymptotic beyond WKB-theory and beyond Maslov canonical operator to the Colombeau solutions of the n-dimensional Schrodinger equation is presented. Quantum jumps nature is considered successfully. We pointed out that an explanation of quantum jumps can be found to result from Colombeau solutions of the Schrodinger equation alone without additional postulates.

On the High-Order Quasi Exactly Solvable Differential Equations

In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.

On Least Squares Solutions of Matrix Equation MZN=S

Let be a given Hermitian matrix satisfying . Using the eigenvalue decomposition of , we consider the least squares solutions to the matrix equation , with the constraint .

Tau-Collocation Approximation Approach for Solving First and Second Order Ordinary Differential Equations

This paper presents Tau-collocation approximation approach for solving first and second orders ordinary differential equations. We use the method in the stimulation of numerical techniques for the approximate solution of linear initial value problems (IVP) in first and second order ordinary differential equations. The resulting numerical evidences show the method is adequate and effective.

Exactly Solvable Schrodinger Equation with Hypergeometric Wavefunctions

In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function;the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices g(x)~P(x)/x2 and g(x)~x2/P(x) give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.

线性正则正余弦加权卷积及其应用

针对积分方程的求解问题,本文提出了利用卷积运算及其卷积定理来讨论两类卷积类积分方程组的解。首先,在线性正则正弦变换与线性正则余弦变换的基础上,定义了线性正则正余弦卷积运算及其加权卷积运算;其次,推导了相应的卷积定理,研究了该卷积与傅里叶正余弦变换卷积运算的关系;最后,讨论了两类卷积类积分方程组的解,给出了该方程解的一般形式。