Zhang Ming-Jiang

Conformal Invariance and a New Type of Conserved Quantities of Mechanical Systems with Variable Mass in Phase Space

Conformal invariance and a new type of conserved quantities of mechanical systems with variable mass in phase space are studied. Firstly, the definition and determining equation of conformal invariance are presented. The relationship between the conformal invariance and the Lie symmetry is given, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry under the infinitesimal transformations is provided. Secondly, a new type of conserved quantities of the conformal invariance are obtained by using the Lie symmetry of the system. Lastly, an example is given to illustrate the application of the results.

Perturbation to Noether Symmetry and Noether Adiabatic Invariants of General Discrete Holonomic Systems

The perturbation to Noether symmetry and Noether adiabatic invariants of general discrete holonomic systems are studied. First, the discrete Noether exact invariant induced directly from the Noether symmetry of the system without perturbation is given. Secondly, the concept of discrete high-order adiabatic invariant is presented, the criterion of the perturbation to Noether symmetry is established, and the discrete Noether adiabatic invariant induced directly from the perturbation to Noether symmetry is obtained. Lastly, an example is discussed to illustrate the application of the results.

Conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems ⁄

This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invariance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.

Perturbation to Lie–Mei Symmetry and Adiabatic Invariants for Birkhoffian Systems

Based on the concept of adiabatic invariant, the perturbation to Lie–Mei symmetry and adiabatic invariants for Birkhoffian systems are studied. The definition of the perturbation to Lie–Mei symmetry for the system is presented, and the criterion of the perturbation to Lie–Mei symmetry is given. Meanwhile, the Hojman adiabatic invariants and the Mei adiabatic invariants for the perturbed system are obtained.

A New Type of Conserved Quantity of Mei Symmetry for Holonomic Mechanical System

A new type of conserved quantity which is directly induced by the Mei symmetry of the holonomic system is studied. Firstly, the definition and criterion of the Mei symmetry for a holonomic mechanical system is given. Secondly, the condition of existence of the new conserved quantity as well as its form is obtained. Lastly, an example is given to illustrate the application of the results.

A New type of conserved quantity deduced from Mei symmetry of nonholonomic systems in terms of quasi-coordinates

This paper studies the new type of conserved quantity which is directly induced by Mei symmetry of nonholonomic systems in terms of quasi-coordinates. A coordination function is introduced, and the conditions for the existence of the new conserved quantities as well as their forms are proposed. Some special cases are given to illustrate the generalized significance of the new type conserved quantity. Finally, an illustrated example is given to show the application of the nonholonomic systems results.

Perturbation to Mei symmetry and Mei adiabatic invariants for mechanical systems in phase space

For a perturbed mechanical system in phase space, considering /dt in the structure equation and process of proof including infinitesimal parameter obviously, this paper studies the perturbation to Mei symmetry and adiabatic invariants. Firstly, the exact invariant induced directly from the Mei symmetry of the system without perturbation is given. Secondly, based on the concept of high-order adiabatic invariant, the determining equations of the perturbation to Mei symmetry are established, the condition of existence of the Mei adiabatic invariant led by the perturbation to Mei symmetry is obtained, and its form is presented. Lastly, an example is given to illustrate the application of the results.

New Type of Conserved Quantities of Lie Symmetry for Nonholonomic Mechanical Systems in Phase Space

The new types of conserved quantities, which are directly induced by Lie symmetry of nonholonomic mechanical systems in phase space, are studied. Firstly, the criterion of the weak Lie symmetry and the strong Lie symmetry are given. Secondly, the conditions of existence of the new type of conserved quantities induced by the weak Lie symmetry and the strong Lie symmetry directly are obtained, and their form is presented. Finally, an Appell–Hamel example is discussed to further illustrate the applications of the results.

Perturbation to Unified Symmetry and Adiabatic Invariants for Relativistic Hamilton Systems

Based on the concept of adiabatic invariant, the perturbation to unified symmetry and adiabatic invariants for relativistic Hamilton systems are studied. The definition of the perturbation to unified symmetry for the system is presented, and the criterion of the perturbation to unified symmetry is given. Meanwhile, the Noether adiabatic invariants, the generalized Hojman adiabatic invariants, and the Mei adiabatic invariants for the perturbed system are obtained.

Perturbation to Mei Symmetry and Generalized Mei Adiabatic Invariants for Nonholonomic Systems in Terms of Quasi-Coordinates

By introducing the coordination function f, the generalized Mei conserved quantities for the nonholonomic systems in terms of quasi-coordinates are given. Then based on the concept of adiabatic invariant, the perturbation to Mei symmetry and the generalized Mei adiabatic invariants for nonholonomic systems in terms of quasi-coordinates are studied.