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.

Conformal invariance and conserved quantity of Hamilton systems

Conformal invariance and conserved quantities of general holonomic systems are studied. A one-parameter infinitesimal transformation group and its infinitesimal transformation vector of generators are described. The definition of conformal invariance and determining equation for the system are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition, that conformal invariance of the system would be Lie symmetry, is obtained under the infinitesimal one-parameter transformation group. The corresponding conserved quantity is derived with the aid of a structure equation. Lastly, an example is given to demonstrate the application of the result.

Conformal Invariance and Conserved Quantities of General Holonomic Systems

Based on the invariance of differential equations under infinitesimal transformations, Lie symmetry, laws of conservations, perturbation to the symmetries and adiabatic invariants of Poincare equations are presented. The concepts of Lie symmetry and higher order adiabatic invariants of Poincare equations are proposed. The conditions for existence of the exact invariants and adiabatic invariants are proved and their forms are also given. In addition, an example is presented to illustrate these results.

Lie symmetries, perturbation to symmetries and adiabatic invariants of Poincaré equations

The conformal invariance and conserved quantities of Mei symmetry for the non-holonomic systems of non-Chetaev’s type are studied Firstly, the differential equations of motion of the holonomic systems corresponding to the non-holonomic systems are discussed Secondly, based on the introduction of an infinitesimal transformation group and its infinitesimal transformation vector of generators, the definition of conformal invariance and determining equation of Mei symmetry of the systems are provided Thirdly, using restriction equations and additional restriction equations, the conformal invariance of weak and strong Mei symmetry for the systems is given, and the system’s corresponding conserved quantities are obtained, according to the structure equation satisfied by the gauge function At the end of the paper, an example is given to illustrate the application of the result

Conformal invariance of Mei symmetry for the non-holonomic systems of non-Chetaev’s type

Perturbation to Noether symmetries and adiabatic invariants of discrete nonholonomic nonconservative mechanical systems on an uniform lattice are investigated. Firstly, we review Noether symmetry and conservation laws of a nonholonomic nonconservative system. Secondly, we study continuous Noether symmetry of a discrete nonholonomic system, give the Noether symmetry criterion and theorem of discrete corresponding holonomic system and nonholonomic system. Thirdly, we study perturbation to Noether symmetry of the discrete nonholonomic nonconservative system, give the criterion of perturbation to Noether symmetry for this system, and based on the definition of adiabatic invariants, we construct the theorem under which can lead to Noether adiabatic invariants for this system, and the forms of discrete Noether adiabatic invariants are given. Finally, we give an example to illustrate our results.

Perturbation to symmetry and adiabatic invariants of discrete nonholonomic nonconservative mechanical system

The Lie symmetries and conserved quantities of constrained mechanical systems are studied. Using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the systems are established. The structure equation and the form of conserved quantities are obtained. We find the corresponding conserved quantity from a known Lie symmetry, that is a direct problem of the Lie symmetries. And then, the inverse problem of the Lie symmetries-finding the corresponding Lie symmetry from a known conserved quantity-is studied. Finally, the relation between the Lie symmetry and the Noether symmetry is given.

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Symmetries and conserved quantities of constrained mechanical systems

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Symmetries and conserved quantities of constrained mechanical systems