In this paper, the perturbation problem of Lie symmetries and adiabatic invariants for variable mass systems with unilateral holonomic constraints are studied. Firstly, a type of generalized Hojman conserved quantity under general infinitesimal transformation is obtained. Then, based on the definition of high-order adiabatic invariants of a mechanical system, the perburbation of Lie symmetries for variable mass systems with unilateral holonomic constraints under small disturbance is discussed and a type of generalized Hojman adiabatic invariants are given. At last, an example to illustrate the application of the results is given.
Another kind of conserved quantity deduced from Mei symmetry of mechanico-electrical system is studied. Under the infinitesimal transformation of groups, another kind of conserved quantity of Mei symmetry of mechanico-electrical system is obtained from the definition and the criterion of Mei symmetry of mechanico-electrical system. Finally, an example is given to illustrate the application of the result.
Based on the total time derivative along the trajectory of the system, the definition and the criterion of Noether form invariance of nonholonomic controllable mechanical systems of non-Chetaev’s type are presented. A new conserved quantity, as well as the Noether conserved quantity are deduced from the Noether-form invariance. An example is given to illustrate the application of the result.
This paper studies the perturbation to symmetries and the adiabatic invariant for nonholonomic controllable mechanical system in the phase space. The exact invariants introduced by the form invariance of the nonholonomic controllable mechanical system in phase space without perturbation are given. Based on the definition of high-order adiabatic invariants of the mechanical system, the perturbation to symmetries and the adiabatic invariant for nonholonomic controllable mechanical system in phase space under the action of small disturbance is investigated, then the form of high-order adiabatic invariants and the conditions for their existence are presented. An example is finally given to illustrate the application of the results.
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