GIBBS-APPELL’S EQUATIONS OF VARIABLE MASS NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Gibbs-Appell’s equations of motion are extended to the most generalvariable mass nonholonomic mechanical systems.Then the Gibbs-Appell’s equations ofmotion in terms of generalized coordinates or quasi-coordinates and an integral variationalprinciple of variable mass nonlinear nonholonomic mechanical systems are obtained.Finally,an example is given.     

EXTENSION OF WHITTAKER EQUATIONS TO NON-HOLONOMIC MECHANICAL SYSTEMS

In 1904, using the energy integral Whittaker studied the reduction of a dynamical problem to a problem with fewer degrees of freedom for the holonomic conservative systems and obtained the Whittaker equation. In this article, Whittaker equations are extended to nonholonomic systems and the generalized Whittaker equations are obtained. And then these equations are transformed into Nielsen's form. Finally an example is given.     

KANE’S EQUATIONS FOR PERCUSSION MOTION OF VARIABLE MASS NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Kane’s equations for the Routh’s form of variable massnonholonomic systems are established.and the Kane’s equations for percussion motionof variable mass holonomic and nonholonomic systems are deduced from them. Secondly,the equivalence to Lagrange’s equations for percussion motion and Kane’sequations is obtained,and the application of the new equation is illustrated by anexample.     

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

ON THE NEW FORMS OF THE DIFFERENTIAL EQUATIONS OF THE SYSTEMS WITH HIGHER-ORDER NONHOLONOMIC CONSTRAINTS

In this paper,the new forms of the differential equations of motion of the systems withhigher-order nonholonomic constraints are obtained at first,and then the equivalencebetween these equations and the known equations is demonstrated.Finally an example isgiven to illustrate the application of our new equations.     

The method of variation on parameters for integration of a generalized Birkhoffian system

This paper focuses on studying the integration method of a generalized Birkhoffian system.The method of variation on parameters for the dynamical equations of a generalized Birkhoffian system is presented.The procedure for solving the problem can be divided into two steps:the first step,a system of auxiliary equations is constructed and its general solution is given;the second step,the parameters are varied,and the solution of the problem is obtained by using the properties of generalized canonical transformation.The method of variation on parameters for the generalized Birkhoffian system is of universal significance,and we take a nonholonomic system and a nonconservative system as examples to illustrate the application of the results of this paper.     

Gauss white noise perturbations of nonholonomic mechanical systems

The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper.It is proved that the differential equations of the first-order moments of the solution process coincide with the corresponding equations in the non-perturbational case,and that there areε~2-terms but noε-terms in the differential equations of the second-order moments.Two propositions are obtained.Finally,an example is given to illustrate the application of the results.     

Lie symmetries and conserved quantities of nonconservative nonholonomic systems in phase space

The invariance and conserved quantities of the nonconservative nonholonomic systems are studied by introducing the infinitesimal transformations in phase space.The Lie’s symmetrical determining equations are establish ed.The Lie’s symmetrical structure equation is obtained.An example to illustrate the application of the result is given.     

Extension of Whittaker equations to non-holonomic mechanical systems

In 1904, using the energy integral Whittaker studied the reduction of a dynamical problem to a problem with fewer degrees of freedom for the holonomic conservative systems and obtained the Whittaker equation^[1].In this article, Whittaker equations are extended to non-holonomic systems and the generalized Whittaker equations are obtained. And then these equations are transformed into Kiel-sen’s form.Finally an example is given.     

Form invariance and conserved quantity for weakly nonholonomic system

The form invariance and the conserved quantity for a weakly nonholonomic system （WNS） are studied. The WNS is a nonholonomic system （NS） whose constraint equations contain a small parameter. The differential equations of motion of the system are established. The definition and the criterion of form invariance of the system are given. The conserved quantity deduced from the form invariance is obtained. Finally, an illustrative example is shown.     

APPLICABILITY OF HAMILTON-JACOBI METHOD TO NONLINEAR NONHOLONOMIC SYSTEMS

This paper uses Poincare formalism to obtain a generalization of the Hamilton-Jacobi method of integrating dynamical systems moving with nonlinear nonholonomic constraints. Necessary and sufficient conditions are investigated for the applicability of this method to such systems. The method is illustrated by considering some concrete examples of nonholonomic systems.     

The activation method for discretized conservative nonlinear stability problems with multiple parameter and state variables

For nonlinear stability problems of discretized conservative systems with multiple parameter variables and multiple state variables,the activation method is put forward,by which activated potential functions and activated equilibrium equations are derived.The activation method is the improvement and enhancement of Liapunov-Schmidt method in elastic stability theory.It is more generalized and more normalized than conventional perturbation methods.The activated potential functions may be transformed into normalized catastrophe potential functions.The activated equilibrium equations may be treated as bifurcation equations.The researches in this paper will motivate the combination of elastic stability theory with catastrophe theory and bifurcation theory.     

GENERALIZED NOETHER’S THEOREM OF NONHOLONOMIC NONPOTENTIAL SYSTEM IN NONINERTIAL REFERENCE FRAMES

The new Lagrangian of the relative motion of mechanical system is constructed,thevariational principles of Jourdain’s form of nonlinear nonholonomic nonpotential system innoninertial reference frame are established,the generalized Noether’s theorem of thesystem above is presented and proved,and the conserved quantities of system are studied.     

Lie symmetries and conserved quantities of second-order nonholonomic mechanical system

The Lie symmetries and the conserved quantities of the second-order nonholonomic mechanical system are studied. Firstly, by using the invariance of the differential equation of motion under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the system are established, and the structure equation and the conservative quantities of the Lie symmetries are obtained. Secondly , the inverse problems of the Lie symmetries are studied . Finally , an example is given to illustrate the application of the result.     

Formulation and solution for inverse problem of nonholonomic dynamics

This paper presents a formulation and solution for the inverse problem ofnonholonomic dynamics:to find the form of nonholonomic constraints when someintegrals are given and to find the generalized reactive forces of constraint acting onthe system when the expression of the kinetic energy is given.An example is given toillustrate the application of the result.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR GENERALIZED DRINFELD-SOKOLOV EQUATIONS

Ansatz method and the theory of dynamical systems are used to study the traveling wave solutions for the generalized Drinfeld-Sokolov equations. Under two groups of the parametric conditions, more solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions are obtained. Exact explicit parametric representations of these travelling wave solutions are given.     

Generalized variational principles for boundary value problem of electromagnetic field in electrodynamics

An expression of the generalized principle of virtual work for the boundary value problem of the linear and anisotropic electromagnetic field is given. Using Chien＇s method, a pair of generalized variational principles （GVPs） are established, which directly leads to all four Maxwell＇s equations, two intensity-potential equations, two constitutive equations, and eight boundary conditions. A family of constrained variational principles is derived sequentially. As additional verifications, two degenerated forms are obtained, equivalent to two known variational principles. Two modified GVPs are given to provide the hybrid finite element models for the present problem.     

EFFECT OF TEMPERATURE-DEPENDENT PROPERTIES ON THERMOELASTIC PROBLEMS WITH THERMAL RELAXATIONS

Based on the generalized thermoelasticity proposed by Green and Lindsay, the dynamic response of generalized thermoelastic problems with temperature-dependent material properties is investigated. The governing equations are formulated and found to be nonlinear because of the temperature-dependence of properties. Owing to the nonlinearity of the governing equations, the finite element method is resorted to for solution. The results obtained show that the temperature-dependent properties influence the variables considered by reducing their magnitudes. This indicates that taking the temperature-dependence of properties into consideration in the investigation of generalized thermoelastic problems is necessary and practical for accurately predicting the thermoelastic behavior.