Basic theory of fractional Mei symmetrical perturbation and its applications

2020

Complexity

The time-scale dynamic equations play an important role in modeling complex dynamical processes. In this paper, the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations are studied. The definition and criterion of the Mei symmetry of the Birkhoffian system on time scales are given. The conditions and forms of new conserved quantities which are found from the Mei symmetry of the system are derived. As a special case, the Mei symmetry of time-scale Hamilton canonical equations is discussed and new conserved quantities for the Hamiltonian system on time scales are derived. Two examples are given to illustrate the application of results.

Section: Theorem 3 If the Infinitesimals ξ 0 And ξ υ Of The Mei Symmmentioning

confidence: 99%

“…is method has been successfully applied in equations of motion for Lagrangian systems, Hamiltonian systems, Birkhoffian systems, the motion of charged particles in an electromagnetic field, the equation of nonmaterial volumes, the equation of thin elastic rod, etc. [13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Mei Symmetry and New Conserved Quantities of Time-Scale Birkhoff’s Equations

Zhai

2020

Complexity

“…From Mei symmetry, a new kind of conservation law can be brought about, which is different from the Noether or Hojman one and called the Mei conserved quantity. The Mei symmetry theorem has been extended to fractional-order mechanical systems [18,19] and nonstan-dard Lagrangian dynamics [20]. Recently, we studied Mei symmetries on time scales [21,22], but the research is preliminary and limited to Lagrange equations and Birkhoff equations.…”

Section: Introductionmentioning

confidence: 99%

Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

Advances in Mathematical Physics

2021

The Mei symmetry and conservation laws for time-scale nonshifted Hamilton equations are explored, and the Mei symmetry theorem is presented and proved. Firstly, the time-scale Hamilton principle is established and extended to the nonconservative case. Based on the Hamilton principles, the dynamic equations of time-scale nonshifted constrained mechanical systems are derived. Secondly, for the time-scale nonshifted Hamilton equations, the definitions of Mei symmetry and their criterion equations are given. Thirdly, Mei symmetry theorems are proved, and the Mei-type conservation laws in time-scale phase space are driven. Two examples show the validity of the results.

“…There is also a close relationship between these symmetries. Many important results were obtained by Mei and other workers [7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Mei Symmetry of Time-Scales Euler-Lagrange Equations and Its Relation to Noether Symmetry

Zhai

2019

Acta Phys. Pol. A

In this paper, the Mei symmetry of the Euler-Lagrange equations on time-scales and its relation to the Noether symmetry are investigated. The definition and criterion of Mei symmetry of the Lagrangian system on time-scales are given. The conditions and forms of new conserved quantities which are found from the Mei symmetry of the system are derived. In addition, the Noether symmetry of a variational problem for Lagrangian on time-scales under the action of infinitesimal generator vectors and its corresponding conserved quantity are discussed. The results show that the Euler-Lagrange equations on time-scales, the Noether identity and the Noether conserved quantity of the variational problem under discussion are the same with the criterion equations, the structural equation, and the conserved quantity of the Mei symmetry for the original Lagrangian system on time-scales, respectively. In the end, two examples are provided to illustrate applications of the results.