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The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamilton principle for nonholonomic systems on time scales is established, and the corresponding generalized Chaplygin equations are deduced. The reduced Chaplygin equations are also presented. Two special cases of the generalized Chaplygin equations on time scales, where the time scales are equal to the set of real numbers and the integer set, are discussed. Finally, several examples are given to illustrate the application of the results.
The calculus on time scales, which is initiated by Hilger in his PhD thesis in 1988,[1] can be used to develop the theory of dynamic equations on time scales in order to unify and extend the usual differential equations and difference equations. The similarities and differences between continuous and discrete systems are revealed. The physical essences of the continuous and discrete systems and the other complex dynamical systems can be expressed more clearly and accurately by the theory on time scales. Therefore, the theories on time scales have been used in many fields such as mathematics, mechanics, economics, etc.[2–11]
The variational problems on time scales were first introduced by Bohner in 2004.[12] After that the theory on time scales was applied to economics by Atici, Biles and Lebedinsky,[7] the Euler–Lagrange equations of variational problems with nabla derivatives were established. Martins and Torres extended the variational problems with delta derivatives to the variational problems with nabla derivatives,[13] and the higher order Euler–Lagrange equations with nabla derivatives were presented. The variational problems with delta and nabla derivatives were further discussed by Torres,[14] and the paper proved that the unification and extension of variational on time scales was not unique. Besides, Bartosiewicz, Martins and Torres presented the second Euler–Lagrange equations on time scales.[15] Moreover, the theories on time scales were extended to optimal control systems by Malinowska and Ammi,[16] and the corresponding Euler–Lagrange equations were derived.
As is well known, the holonomic systems are described by the second Lagrange equations, while the nonholonomic systems are described by more complex differential equations of motion. However, the controversy exists on the exchange relations for nonholonomic systems.[17–19] Historically, there were two views on the form of the exchange relations for nonholonomic systems: one is the Hölder view, the other is the Suslov view.
Recently, the differential equations of motion with multipliers for nonholonomic system on time scales were established by Fu et al.[20] Zu and Zhu[21] deduced the Hamilton canonical equations for the nonholonomic system in phase space on time scales. Then, Song and Zhang extended the variational problem for mechanical systems on time scales to Birkhoffian systems.[22] Furthermore, the methods of reduction for a Lagrange system with nabla derivative were discussed by Jin and Zhang.[23] Plenty of important results concerning the differential equations on time scales have been obtained.[24–28] However, the differential equations of motion for Chaplygin systems on time scales have not been investigated yet in the literature.
In this paper, we mainly establish the generalized Chaplygin equations according to the Suslov view on time scales. The rest of this paper is organized as follows. In section
We mainly discuss the generalized Chaplygin equations according to the Suslov view on time scales in this section. The basic definitions and facts concerning the calculus on time scales used in the following section can be found in Refs. [4] and [5].
Assume that the configuration of a mechanical system is determined by n generalized coordinates qk
From Eq. (
If
If
The general Chaplygin equations (
In this section, we present the Chaplygin equations on time scales. The generalized Chaplygin equations (
In general, if neither the cycle coordinates
If the constraints of the nonholonomic system on time scales are subjected to first order linear homogeneous stationarity, and the coordinates
Equation (
If
If
In general, the Chaplygin equations (
Time scales are an arbitrary nonempty closed subset of the set of real numbers. The study of differential equations on time scales, which has been created in order to unify and extend the study of differential and difference equations, is an area of science and engineering, which has recently gained a lot of attention.
The paper presents the generalized Chaplygin equations on time scales. The Hamilton principles for nonholonomic systems on time scales are established, and the corresponding generalized Chaplygin equation (see Eq. (
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