This paper discusses the symmetry of Lagrangians of holonomic systems in terms of quasi-coordinates.Firstly,the definition and the criterion of the symmetry are given.Secondly,the condition under which there exists a conserved quantity and the form of the conserved quantity are obtained.Finally,an example is shown to illustrate the application of the results.
This paper presents a Poisson theory of the generalized Birkhoff equations,including the algebraic structure of the equations,the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.
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.
This paper is intended to apply the potential integration method to the differential equations of the Birkhoffian system.The method is that,for a given Birkhoffian system,its differential equations are first rewritten as 2n first-order differential equations.Secondly,the corresponding partial differential equations are obtained by potential integration method and the solution is expressed as a complete integral.Finally,the integral of the system is obtained.
This paper studies the Hojman conserved quantity,a non-Noether conserved quantity,deduced by special weak Noether symmetry for Lagrange systems.Under special infinitesimal transformations in which the time is not variable,its criterion is given and a method of how to seek the Hojman conserved quantity is presented.A Hojman conserved quantity can be found by using the special weak Noether symmetry.
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.
Based on the weak Noether symmetry proposed by Mei F X,this paper discusses the weak Noether symmetry for nonholonomic system of non-Chetaev type,and presents expressions of three kinds of conserved quantities by weak Noether symmetry.Finally,the application of this new results is showed by a practical example.
