Mei Feng-Xiang

Form invariance of appell equations

The form invariance of Appell equations of holonomic mechanical systems under the infinitesimal transformations of groups is studied. The definition and the criterion of the form invariance of Appell equations are given. This form invariance can lead to a conserved quantity under certain conditions.

Conformal invariance and conserved quantity of Hamilton systems

This paper studies conformal invariance and conserved quantities of Hamilton system. The definition and the determining equation of conformal invariance for Hamilton system are provided. The relationship between the conformal invariance and the Lie symmetry are discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal one-parameter transformation group is deduced. It gives the conserved quantities of the system and an example for illustration.

Perturbation to the symmetries and adiabatic invariants of holonomic variable mass systems

The perturbation problem of symmetries for the holonomic variable mass systems under small excitation is discussed. The concept of high-order adiabatic invariant is presented, and the form of adiabatic invariants and the conditions for their existence are given. Then the corresponding inverse problem is studied. Finally an example is presented to illustrate these results.

SYMMETRIES AND CONSERVED QUANTITIES FOR SYSTEMS OF GENERALIZED CLASSICAL MECHANICS

In this paper, the symmetries and the conserved quantities for systems of generalized classical mechanics are studied. First, the generalized Noether’s theorem and the generalized Noether’s inverse theorem of the systems are given, which are based upon the invariant properties of the canonical action with respect to the action of the infinitesimal transformation of r-parameter finite group of transformation; second, the Lie symmetries and conserved quantities of the systems are studied in accordance with the Lie’s theory of the invariance of differential equations under the transformation of infinitesimal groups; and finally, the inner connection between the two kinds of symmetries of systems is discussed. PACC: 0320; 1110; 0220 I. INTRODUCTION The conservation laws (or first integrals) of the mechanical systems are always of mathematical importance and at the same time, they are regarded as the manifestation of some profound physical principles. The fact that the existence of a conserved quantity means that an inner dynamical symmetry in the mechanical system plays an irreplaceable role in the physical interpretation of motion. The modern methods in finding conservation laws are mainly two ways [1] with each having a concept of its own: the first one is based upon the invariant properties of Hamiltonian action with respect to the action of the infinitesimal transformation of the finite transformation group, i.e., Noether symmetries and conserved quantities; the second one is based upon the Lie’s theory of the invariance of differential equations under the action of infinitesimal groups. The differential equations of many physical problems appear to be the variational problems. The theory of the generalized classical mechanics, initiated by Ostrogradsky and Jacobi in the years from 1848 to 1858, has been gaining much development during the recent fifty years, and many important results have been made. [2 6] The Lagrange equations with high

Conformal invariance and integration of first-order differential equations

This paper studies a conformal invariance and an integration of first-order differential equations. It obtains the corresponding infinitesimal generators of conformal invariance by using the symmetry of the differential equations, and expresses the differential equations by the equations of a Birkhoff system or a generalized Birkhoff system. If the infinitesimal generators are those of a Noether symmetry, the conserved quantity can be obtained by using the Noether theory of the Birkhoff system or the generalized Birkhoff system.

Form invariance for systems of generalized classical mechanics

This paper presents a form invariance of canonical equations for systems of generalized classical mechanics. According to the invariance of the form of differential equations of motion under the infinitesimal transformations, this paper gives the definition and criterion of the form invariance for generalized classical mechanical systems and establishes relations between form invariance, Noether symmetry and Lie symmetry. At the end of the paper, an example is given to illustrate the application of the results.

Form invariance and Lie symmetry of equations of non-holonomic systems

In this paper, we study the relation between the form invariance and Lie symmetry of non-holonomic systems. Firstly, we give the definitions and criteria of the form invariance and Lie symmetry in the systems. Next, their relation is deduced. We show that the structure equation and conserved quantity of the form invariance and Lie symmetry of non-holonomic systems have the same form. Finally, we give an example to illustrate the application of the result.

Noether's theory of mechanical systems with unilateral constraints

Noethers theory of dynamical systems with unilateral constraints by introducing the generalized quasi-symmetry of the infinitesimal transformation for the transformation group G, is presented and two examples to illustrate the application of the result are given.

Stability for manifolds of equilibrium states of generalized Birkhoff system

Stability for the manifolds of equilibrium states of a generalized Birkhoff system is studied. A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to the generalized Birkhoffian system and two propositions on the stability of the manifolds of equilibrium states of the system are obtained. An example is given to illustrate the application of the results.

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 with higher-order nonholonomic constraints are obtained at first, and then the equivalence between these equations and the known equations is demonstrated. Finally an example is given to illustrate the application of our new equations.