Adiabatic invariants and Lie symmetries on time scales for nonholonomic systems of non-Chetaev type

Abstract

Time-scale dynamics integrates the differential equations of continuous systems and the difference equations of discrete systems. It can not only reveal the similarities and differences between continuous and discrete systems, but also describe the physical nature of continuous and discrete systems and other complex dynamical systems more clearly and accurately. Therefore, it has been widely used in many fields of science and engineering in recent years. In this paper, we investigate Lie symmetries and invariants of nonholonomic systems of non-Chetaev type on time scales. First, we present and prove the Lie symmetry theorem for undisturbed nonholonomic systems of non-Chetaev type on time scales. The study shows that if the Lie symmetry satisfies the structural equation, it will lead to the conserved quantity, which is the exact invariant of the system. Secondly, considering that the system is subjected to small disturbance, we present and prove the adiabatic invariant theorem of Lie symmetry for nonholonomic systems of non-Chetaev type on time scales. Due to the arbitrariness of the time scale, the method and results of this paper are of universal significance. An example is given to illustrate the validity of the results.