Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems

Abstract

For a generalized Birkhoffian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants are presented. On the basis of the invariance of disturbed generalized Birkhoffian system under general infinitesimal transformation of group, the determining equation of Lie symmetrical perturbation of the system is constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of non-Noether adiabatic invariants of a disturbed generalized Birkhoffian system is obtained by investigating the Lie symmetrical perturbation. Then, a new type of exact invariants of nonNoether type is given, furthermore adiabatic invariants and exact invariants of non-Noether type are obtained under the special infinitesimal transformation of group. Finally, an example is given to illustrate the application of the method and results.

Fractional Birkhoffian mechanics

“…The Birkhoffian dynamics is more general than the Hamiltonian dynamics. The Hamiltonian dynamics has been extensively applied in many fields of science and engineering, so the Birkhoffian dynamics should also play a more important role in these fields and has been applied to nonlinear dynamical systems [17][18][19][20][21], relativistic systems [22], rotational relativistic systems [23,24] and quantum systems [25], etc.…”

Conformal Invariance and Conserved Quantity of the Higher-Order Holonomic Systems by Lie Point Transformation

“…Thus, in recent years, to seek conserved quantities using symmetries of dynamic systems has been a modern development direction, and some important results have been gained so far [2][3][4][5][6]. Symmetry methods mainly include Noether theory [7][8][9], Lie symmetry [10][11][12][13][14][15][16] and form invariance [17], i.e. Mei symmetry [18][19][20][21], and so on [22,23].…”

Lie symmetry and approximate Hojman conserved quantity of Appell equations for a weakly nonholonomic system

“…In the recent 20 years, Chinese researchers have gained fruitful achievements in research, promotion, and application of Appell equations [12][13][14][15][16][17]. Since 2000, Chinese researchers have made some achievements in this research area, especially in Lie symmetry [35][36][37][38][39][40][41][42][43][44][45][46][47] of constrained mechanical systems [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. However, there are fewer results on Appell equations.…”