Lie symmetries, symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems

Abstract

For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries, symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations, and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations, constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained, and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of nonNoether exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is given to illustrate the application of the method and results.

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

“…For the generalized Birkhoffian system (3) without perturbations, if we set τ −1 = 0, ξ −1 μ = 0 and λ = λ 0 when m = 0, then Theorem 1 gives the Lie symmetrical exact invariants of a new non-Noether type for the generalized Birkhoffian system. Theorem 2 For the generalized Birkhoffian system (3) without perturbations, if the generators τ 0 , ξ 0 μ of the infinitesimal transformations (7) satisfy the Lie symmetrical determining (16), and there exists a function λ 0 = λ 0 (t, a μ ) satisfying the condition…”

“…For a mechanical system, there exists an intimate relation between the integrability of the system and the variations of its symmetries and invariants under the action of small disturbance and, therefore, the researches on symmetrical perturbation and adiabatic invariants are of great significance. So, more and more attention has been paid to research on symmetrical perturbation and adiabatic invariants of the mechanical system; some important results have been reported in recent years, and the study of symmetrical perturbation and adiabatic invariants has become a popular subject in mechanics, relativistic systems, atomic and molecular physics, spherical physics, and engineering [5][6][7][8][9][10][11][12][13][14][15][16].…”

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.…”

Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

“…In 1997, Galiullin et al had proposed concepts of conformal invariance and conformal factor of Birkhoff equations in the study of dynamics in Birkhoffian systems [23], and discussed the relationships between the invariance and the conformal invariance, Lie symmetry and the conformal invariance under the infinitesimal transformations of Pfaff action. Since entering the twenty-first century, Chinese scholars have carried out certain new researches and have gained some achievements in the study of symmetries and conserved quantities for mechanical systems by means of theories of the conformal invariance [24][25][26][27][28][29][30][31][32][33][34][35]. However, for a long time, there are fewer results to Appell equations.…”