Non-Noether Conserved Quantity for Relativistic Nonholonomic System with Variable Mass

Using form invariance under special infinitesimal transformations in which time is not variable, the non-Noether conserved quantity of the relativistic nonholonomic system with variable mass is studied. The differential equations of motion of the system are established. The definition and criterion of the form invariance of the system under infinitesimal transformations are studied. The necessary and sufficient. condition under which the form invariance is a Lie symmetry is given. The condition under which the form invariance can be led to a non-Noether. conserved quantity and the form of the conserved quantity are obtained. Finally, an example is given to illustrate the application of the result.

Non-Noether Conserved Quantity of Poincaré-Chetaev Equations of a Generalized Classical Mechanics

In the present paper the Lie symmetrical non-Noether conserved quantity of the Poincaré Chetaev equations of a generalized classical mechanics under the general infinitesimal transformations of Lie groups is discussed. First, we establish the determining equations of Lie symmetry of the equations. Second, the Lie symmetrical non-Noether conserved quantity of the equations is deduced. Finally, an example is given to illustrate the application of the results.

Noether Symmetry Can Lead to Non-Noether Conserved Quantity of Holonomic Nonconservative Systems in General Lie Transformations

For the holonomic nonconservative system, by using the Noether symmetry, a non-Noether conserved quantity is obtained directly under general infinitesimal transformations of groups in which time is variable. At first,the Noether symmetry, Lie symmetry, and Noether conserved quantity are given. Secondly, the condition under which the Noether symmetry is a Lie symmetry under general infinitesimal transformations is obtained. Finally, a set of nonNoether conserved quantities of the system are given by the Noether symmetry, and an example is given to illustrate the application of the results.

Lie Symmetry and Non-Noether Conserved Quantity for Hamiltonian Systems

A non-Noether conserved quantity for the Hamiltonian system is studied. A particular infinitesimal transformation is given and the determining equations of Lie symmetry are established. An existence theorem of the non-Noether conserved quantity is obtained. An example is given to illustrate the application of the result.

New non-Noether conserved quantities of mechanical system in phase space

This paper focuses on studying non-Noether conserved quantities of Lie symmetry and of form invariance for a mechanical system in phase space under the general infinitesimal transformation of groups. We obtain a new nonNoether conserved quantity of Lie symmetry of the system, and Hojman and Mei＇s results are of special cases of our con-clusion. We find a condition under which the form invariance of the system will lead to a Lie symmetry, and, further, obtain a new non-Noether conserved quantity of form invariance of the system. An example is given finally to illustrate these results.

Weak Noether Symmetry and non-Noether Conserved Quantities for General Holonomic Systems

A new kind of weak Noether symmetry for a general holonomic system is defined in such a way that themethods to construct Hojman conserved quantity and new-type conserved quantity are given.It turns out that weintroduce a new approach to look for the conserved laws.Two examples are presented.

Lie Symmetrical Non-Noether Conserved Quantities of Poincaré-Chetaev Equations

In the present paper the Lie symmetrical non-Noether conserved quantity of the Poincaré-Chetaev equations under the general infinitesimal transformations of Lie groups is discussed. First, we establish the determining equations of Lie symmetry of the equations. Second, the Lie symmetrical non-Noether conserved quantity of the equations is deduced.

Non-Noether conserved quantity for differential equations of motion in the phase space

A non-Noether conserved quantity for the differential equations of motion of mechanical systems in the phase space is studied. The differential equations of motion of the systems are established and the determining equations of Lie symmetry are given. An existence theorem of non-Noether conserved quantity is obtained. An example is given to illustrate the application of the result.

An analogical study of wave equations,physical quantities,conservation and reciprocity equations between electromagnetic and elastic waves

This paper presents an analogical study between electromagnetic and elastic wave fields,with a one-to-one correspondence principle established regarding the basic wave equations,the physical quantities and the differential operations.Using the electromagnetic-to-elastic substitution,the analogous relations of the conservation laws of energy and momentum are investigated between these two physical fields.Moreover,the energy-based and momentum-based reciprocity theorems for an elastic wave are also derived in the time-harmonic state,which describe the interaction between two elastic wave systems from the perspectives of energy and momentum,respectively.The theoretical results obtained in this analysis can not only improve our understanding of the similarities of these two linear systems,but also find potential applications in relevant fields such as medical imaging,non-destructive evaluation,acoustic microscopy,seismology and exploratory geophysics.

A series of non-Noether conservative quantities and Mei symmetries of nonconservative systems

In this paper Mei symmetry is introduced for a nonconservative system. The necessary and sufficient condition for a Mei symmetry to be also a Lie symmetry is derived. It is proved that the Mei symmetry leads to a non-Noether conservative quantity via a Lie symmetry, and deduces a Lutzky conservative quantity via a Lie point symmetry.

Non-Noether symmetries and Lutzky conservative quantities of nonholonomic nonconservative dynamical systems

Non-Noether symmetries and conservative quantities of nonholonomic nonconservative dynamical systems are investigated in this paper. Based on the relationships among motion, nonconservative forces, nonholonomic constrained forces and Lagrangian, non-Noether symmetries and Lutzky conservative quantities are presented for nonholonomic nonconservative dynamical systems. The relation between non-Noether symmetry and Noether symmetry is discussed and it is further shown that non-Noether conservative quantities can be obtained by a complete set of Noether invariants. Finally, an example is given to illustrate these results.

Lie Symmetries and Conserved Quantities for the Singular Lagrange System

The invariance of the ordinary differential equations under the infinitesimal transformations was used to study the Lie symmetries and conserved quantities for the singular Lagrange system. The determining equations, the restriction equations of the Lie symmetries and the form of conserved quantities of the system are obtained.

Conserved quantities from Lie symmetries for nonholonomic systems

This paper presents a new method to seek the conserved quantity from a Lie symmetry without using either Lagrangians or Hamiltonians for nonholonomic systems. The differential equations of motion of the systems are established. The definition of the Lie symmetrical transformations of the systems is given, which only depends upon the infinitesimal transformations of groups for the generalized coordinates. The conserved quantity is directly constructed in terms of the Lie symmetry of the systems. The condition under which the Lie symmetry can lead to the conserved quantity and the form of the conserved quantity are obtained. Finally, an example is given to illustrate the application of the result.

Lie Symmetries and Conserved Quantities of Holonomic Mechanical Systems in Terms of Quasi-Coordinatee

Aim To study the Lie symmetries and conserved quantities of holonomic mechanical systems in terms of qnasi-coordinates. Methods The definition of an infinitesimal generator for the holonomic mechanical systems in terms of quasi-coordinates was given. Lie's method of the invariance of ordinary differential equations under infinitesimal transformations was used. Results and Conclusion The determining equations of the Lie symmetries of holonomic mechanical systems in terms of quassi-coordinates are established. The structure equation and the form of conserved quantities are obtained. An example to illustrate the applicaiton of the result is given.

Symmetries and conserved quantities of generalized Birkhoffian systems

Three kinds of symmetries and their corresponding conserved quantities of a generalized Birkhoffian system are studied. First, by using the invariance of the Pfaffian action under the infinitesimal transformations, the Noether theory of the generalized Birkhoffian system is established. Secondly, on the basis of the invariance of differential equations under infinitesimal transformations, the definition and the criterion of the Lie symmetry of the generalized Birkhoffian system are established, and the Hojman conserved quantity directly derived from the Lie symmetry of the system is given. Finally, by using the invariance that the dynamical functions in the differential equations of the motion of mechanical systems still satisfy the equations after undergoing the infinitesimal transformations, the definition and the criterion of the Mei symmetry of the generalized Birkhoffian system are presented, and the Mei conserved quantity directly derived from the Mei symmetry of the system is obtained. Some examples are given to illustrate the application of the results.

Lie Symmetries and Conserved Quantities of Holonomic Systems with Remainder Coordinates

Aim To study the Lie symmetries and the consered quantities of the holonomic systems with remainder coordinates. Methods Using the invariance of the ordinary differential equations under the infinitesimal transformations to establish the determining equations and the restriction equations of the Lie symmetries of the systems. Results and Conclusion the structure equation and the form of conserved quantities were obtained. An example was given to illustrate the application of the result.

Lie Symmetries and Conserved Quantities of Systems of Relative Motion Dynamics

Aim To study the Lie symmetries and conserved quantities of the dynamical equationsof relative motion for holonomic mechanical systems. Methods Lie's method of the invariance of ordinary differential equations under infinitesimal transformations was used. Results and Conclusion The determining equaiton of the Lie symmetries for the dynamical equationS of relative motion is established.The structure quation and the form conserved quantities are obtained. An example iD illustrate the application of the result is given.

Mei Symmetry and Hojman Conserved Quantity of Nonholonomic Controllable Mechanical System

A non-Noether conserved quantity, i.e., Hojman conserved quantity, constructed by using Mei symmetry for the nonholonomic controllable mechanical system, is presented. Under general infinitesimal transformations, the determining equations of the special Mei symmetry, the constrained restriction equations, the additional restriction equations, and the definitions of the weak Mei symmetry and the strong Mei symmetry of the nonholonomic controllable mechanical system are given. The condition under which Mei symmetry is a Lie symmetry is obtained. The form of the Hojman conserved quantity of the corresponding holonomic mechanical system, the weak Hojman conserved quantity and the strong Hojman conserved quantity of the nonholonomie controllable mechanical system are obtained. An example is given to illustrate the application of the results.

A symmetry and a conserved quantity for the Birkhoff system

A symmetry and a conserved quantity of the Birkhoff system are studied. The symmetry is called the Birkhoff symmetry. Its definition and criterion are given in this paper. A conserved quantity can be deduced by using the symmetry. An example is given to illustrate the application of the result.

Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales

This paper focuses on studying the Noether symmetry and the conserved quantity with non-standard Lagrangians, namely exponential Lagrangians and power-law Lagrangians on time scales. Firstly, for each case, the Hamilton prin- ciple based on the action with non-standard Lagrangians on time scales is established, with which the corresponding Euler-Lagrange equation is given. Secondly, according to the invariance of the Hamilton action under the infinitesimal transformation, the Noether theorem for the dynamical system with non-standard Lagrangians on time scales is established. The proof of the theorem consists of two steps. First, it is proved under the infinitesimal transformations of a special one-parameter group without transforming time. Second, utilizing the technique of time-re-parameterization, the Noether theorem in a general form is obtained. The Noether-type conserved quantities with non-standard Lagrangians in both clas- sical and discrete cases are given. Finally, an example in Friedmann-Robertson-Walker spacetime and an example about second order Duffing equation are given to illustrate the application of the results.