Lie symmetries and conserved quantities of rotational relativistic systems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ1979,Ｒ.ＢｅｎｇｔｓｓｏｎａｎｄＳ.Ｆｒａｎｅｎｄｏｒｆａｃｃｕｒａｔｌｙｍｅａｓｕｒｅｄｔｈｅｍａｘｉｍｕｍｖａｌｕｅｓｏｆｔｈｅｓｐｉｎｖｅｌｏｃｉｔｙｏｆ14ｋｉｎｄｓｏｆｎｕｃｌｅｏｎｓ,ａｎｄｔｈｅｒｅｓｕｌｔｓｓｈｏｗｅｄｔｈａｔｔｈｅｍａｘｉｍｕｍｖａｌｕｅｏｆｔｈｅｓｐｉｎｖｅｌｏｃｉｔｙｏｆｏｎｅｎｕｃｌｅｏｎｗａｓｄｉｆｆｅｒｅｎｔｔｏｔｈｏｓｅｏｆｔｈｅｏｔｈｅｒｓ[1].Ｗｉｔｈｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ,…     

The theory of relativistic analytical mechanics of the rotational systems

I'IntroductionThespinningmotionisintrinsicattributeofmicroscopicparticle.In1979,R.BengtssonandS.Frauendorfaccuratlymeasuredthemaximumvolumesofthespinningrotativevelocityof14kindsofnucleons,andtheresultsshowedthatthemaximumvolumesofthespinningrotativevelocityofeachnucleonaredifferent[ll.Withthede\'elopingofmodernscienceandtechnology,moreandmoreexperimentalfactsrelatedtothehighspeedrotationquestions,theEinstein'srelativitytheoryandtherotationaltheoryofclassicalmechanicsarenotsuitabletothesepro…     

One type of integrals for the equations of motion of higher-order nonholonomic systems

This paper presents one type of integrals and its condition of existence for theequations Of motion of higher-order nonholonomic systems,including I-order integral(generalized energy integral),2-order integral and p-order integral(P>2).All of theseintegrals can be constructed by the Lagrangianfunction of the system.Two examples aregiven to illustrate the application of the suggested method.     

Algebraic structure and Poisson method for a weakly nonholonomic system

The algebraic structure and the Poisson method for a weakly nonholonomic system are studied. The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed. The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system. An example is given to illustrate the application of the result.     

Connection of first integrals with particular solutions of the nonsimultaneous variational equations for nonholonomic systems

In this paper, the nonsimultaneous variational equations of the nonholonomic mechanical systems are presented, and their solutions are studied. It is proven that, under some conditions, a particular solution of the nonsimultaneous variational equations can be obtained by using a first integral. At the end of the paper, an example is given to illustrate the applications of the results.     

Canonical forms of Nielsen's and Cenov's dynamical equations

In this paper, with Poincaré's formalism, and an indirect method, the canonical forms of the generalized equations of motion due to Nielsen and Cenov of a holonomic dynamical system in the velocity-phase space and the acceleration-phase space are obtained in terms of the Poincaré parameters This paper was presented at the International Congress of Mathematicians (ICM), 21–29 August, 1990, Kyoto University, Japan.     

Equations of motion for nonholonomic mechanical systems with unilateral constraints

IntroductionThemotionofamechanicalsystemwithunilateralconstraintsismoregeneralthanwithbilateralconstraints1andyetitsinvestigationismoredifficult[1,2].Thesystemswithunilateralconstraintsinvolvetheholonomicmechanicalsystemswithunilateralholonomicconstraints[3～6]andthenonholonondcmeChanicalsystemswithunilateralholonondcconstraintS[7]andthenonholonondcmechanicalsystemswithunilateralnonholonondcconstraintS[2]andsoon.ThispaperstUdiesatypeofmoregeneralsystemswithunilateralconstfaints:nonholonondcs…     

Generalization of a new parametric formulation of mechanics for systems with variable mass

In this paper a new parametric formulation of mechanics, formulated by the author himself and based on the separation of the double role of time (independent variable and a parameter) with the aid of a family of varied paths, is extended to the arbitrary rheonomic systems with variable mass. For such systems d'Alembert–Lagrange's principle, general Hamilton's principle, and the corresponding Hamiltonian formalism are formulated, as well as the energy relations with energy change law. The obtained results are illustrated by a simple, but characteristic example.     

A field method for integrating the equations of motion of nonholonomic controllable systems

This paper presents a field method for integrating the equations of motion of nonholonomic controllable systems. An example is given to illustrate the application of the method.     

An Algebraic Method of Stabilization for a Class of Boundary Control Systems of Parabolic Type

Stabilization of linear parabolic boundary control systems is studied. The boundary control system is composed of a system of linear differential operators (, ) in a bounded domain with the smooth boundary. A significant feature of the paper is that, while being a standard elliptic operator, characterizes the boundary condition partly of the Dirichlet type and partly of the generalized Neumann type. We present a new algebraic approach and show that the stabilization is achieved under the standard controllability and observability conditions associated with the system.     

Geometrical theory of fluid flows and dynamical systems

Various dynamical systems have often common geometrical structures and can be formulated on the basis of Riemannian geometry and Lie group theory, provided that a dynamical system has a group symmetry, namely it is invariant under group transformations, and further that the group manifold is endowed with a Riemannian metric. The basic ideas and tools are described, and their applications are presented for the following five problems: (a) free rotation of a rigid body, which is a well-known system in mechanics and presented as an illustrative example of the geometrical theory; (b) geodesic equation and KdV equation on the group of diffeomorphisms of a circle and its extended group; (c) a self-gravitating system of a finite number of points masses and a geometrical interpretation of chaos of Hénon–Heiles system; (d) geometrical formulation of hydrodynamics of an incompressible ideal fluid on the group of volume preserving diffeomorphisms, where an interpretation of the origin of Riemannian curvatures of the fluid flow is given; (e) geodesic equation on a loop group and the local induction equation for the motion of a vortex filament, where the geodesic equation on its extended group is found to be equivalent to the equation for a vortex filament with an axial flow along it.

It is remarkable that the present geometrical formulations are successful for all the problems considered here and give an insight into the deep background common to the diverse physical systems. Furthermore, the geometrical formulation opens a new approach to various dynamical systems, which is rewarded with new results.     

Numerical methods for solving singular integral equations obtained by fracture mechanical analysis of cracked wedge简

The numerical solutions to the singular integral equations obtained by the fracture mechanical analyses of a cracked wedge under three different conditions are considered. The three considered conditions are: (i) a radial crack on a wedge with a non-finite radius under the traction-traction boundary condition, (ii) a radial crack on a wedge with a finite radius under the traction-traction boundary condition, and (iii) a radial crack on a finite radius wedge under the traction-displacement boundary condition. According to the boundary conditions, the extracted singular integral equations have different forms. Numerical methods are used to solve the obtained coupled singular integral equations, where the Gauss-Legendre and the Gauss-Chebyshev polynomials are used to approximate the responses of the singular integral equations. The results are presented in figures and compared with those obtained by the analytical response. The results show that the obtained Gauss-Chebyshev polynomial response is closer to the analytical response.     

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.     

Equations of motion for mechanical systems: A unified approach

This paper presents a unified framework from which emerge the Lagrange equations, the Gibbs-Appell Equations and the Generalized Inverse Equations for describing the motion of constrained mechanical systems. The unified approach extends the applicability of the first two approaches to systems where the constraints are non-linear functions of the generalized velocities and are not necessarily independent. Furthermore, the approach leads to the Explicit Gibbs-Appell Equations.     

Necessary and sufficient condition for the correct formulation of boundary integral equations of harmonic functions

A necessary and sufficient condition for the correct formulation of boundary integral equations of harmonic functions is established in the present paper. A super-determined problem of harmonic functions is proposed for the first time. Then a necessary and sufficient condition for the existence of solution of the super-determined problem is proved. At the same time, it is a necessary and sufficient condition for the correct formulation of boundary integral equations with direct unknown quantities. A relation between boundary integral equations and variational principles is discovered for the first time. And a one-to-one correspondence between boundary integral equations with direct and indirect unknown quantities is indicated. The concept and route of the present paper can be applied to other boundary value problems possessing variational principles.     

A necessary and sufficient set of boundary integral equations with indirect unknowns for plate bending problem

A boundary integral representation of plane biharmonic function is established rigorously by the method of unanalytical continuation in the present paper. In this representation there are two boundary functions and four constants which bear a one to one correspondence to biharmonic functions. Therefore the set of boundary integral equations with indirect unknowns based on this representation is equivalent to the original differential equation formulation.     

The estimation of solution of the boundary value problem of the systems for quasi-linear ordinary differential equations

This paper deals.with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equations where x,f, y, h, A, B and C all belong to Rm, and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder.     

Asynchronous relaxed iterative methods for solving linear systems of equations

I.AsynchronousRelaxedIterativeMethodsMoreandmorelargeorverylargescaleproblemsofscientificcomputationhavebeenproposedandarebeingproposedinmanyimportantfieldsofscienceandengineering.Manyoftheseproblemsresultinsolvinglargeorverylargelinearalgebraicsystemsofe…     

Numerical solutions of singular integral equations for planar rectangular interfacial crack in three dimensional bimaterials

Stress intensity factors for a three dimensional rectangular interfacial crack were considered using the body force method. In the numerical calculations, unknown body force densities were approximated by the products of the fundamental densities and power series; here the fundamental densities are chosen to express singular stress fields due to an interface crack exactly. The calculation shows that the numerical results are satisfied. The stress intensity factors for a rectangular interface crack were indicated accurately with the varying aspect ratio, and bimaterial parameter.     

Classification and integration of four-dimensional dynamical systems admitting non-linear superposition

The method of integration of dynamical systems admitting non-linear superpositions is applied to four-dimensional non-linear dynamical systems. All four-dimensional dynamical systems admitting non-linear superpositions with four-dimensional Vessiot-Guldberg-Lie algebras are classified into 160 standard forms. The integration method is described and illustrated.