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Introduction
A time scale is an arbitrary nonempty closed subset of the real numbers.The calculus of time scales were initiated by B.Aulbach and S.Hilger [1] in order to create a theory which can unify discrete and continuous analysis. A time scale is a model of time, and the new theory has found important applications in several fields which require simultaneous modeling of discrete and continuous data, in the calculus of variations, control theory, and optimal control [2,3,4,5]. The calculus of variations on time scales was initiated with the presentation of Euler-Lagrange equations on time scales was presented in 2004 [6]. But, Torres put forward the second Euler-Lagrange equations and researched the higher-order calculus of variations on time scales [7,8]. The calculus of variations and control theory are disciplines in which there appears to be many opportunities for application of time scales [9,10,11].
In 1918, Noether proposed famous Noether symmetry theorems which could be used to deal with the invariance of the Hamilton action under the infinitesimal transformations: when a system exhibits a symmetry, then a conservation law can be obtained. Bartosiewicz and Torres showed that there existed a conserved quantity in Lagrangian system for each Noether symmetry [12] on time scales by the technique of time-re-parameterization. Using this technique, Cai and Fu studied the theories of Noether symmetries of the nonconservative and nonholonomic systems on time scales [13,14]. Noether theory of the Hamilton systems on time scales was given by Zhang [15,16]. It is worth mentioning that the dynamic systems on time scales with delta derivative have just started to originate.
With the development of modern science and technology, people pay attention to the dynamic of relative motion. Jet aircrafts, rockets, satellite, spacecraft and so on generally involve application of the relative motion systems. We also know that the movement of Mechanical systems is researched in either absolute coordinate system or moving coordinate system. The dynamic systems in the moving coordinate system is called the relative motion dynamic. In 1961, Lur’e introduced the equation of the relative motion systems for conservative systems [17]. In 1993, The Lagrange equation of relative motion dynamics for the general holonomic system was first studied by Liu [18]. In recent decades, a series of innovative research results about dynamics of relative motion have been obtained [19,20,22].
In this letter, we study the Noether symmetry of relative motion systems on time scales. The structure of this letter as follows: In Section 2, we review preparatory knowledge and properties of time scales. In Section 3, we establish the equations of the relative motion systems with delta derivatives. In Section 4, Noether theorems and conserved quantities for the relative motion systems are founded. Lastly, an example is used to illustrate the results.
Previous results of time scales
To begin with, we briefly present some main definitions and properties about times scales. More detailed theory of time scales can refer to [23,24,25,].
Definition 1
A time scale T is an arbitrary nonempty closed subset of the set R of real number. For t ∈ T, we define the forward jump operator σ : T → T by
The graininess function μ : T → [0, ∞] is defined by μ(t) = σ(t) − t for each t ∈ T.
A point is called right-dense, right-scattered, left-dense or left-scattered if σ(t) = t, σ(t) > t, ρ(t) = t, ρ(t) < t, respectively. We say Tk = T − {M} if T has a left-scattered maximum M, otherwise Tk = T.
Definition 2
Assume f : T → R is a function and t ∈ Tk, we define fΔ(t) to be the real number with the property with given any ε, there is neighborhood U = (t − δ, t + δ) ⋂ T of such that
Assume f : T → R is a regulated function, existing a function F with FΔ(t) = f(t) is called a pre-antiderivative of f and in this case an integral of f from a to b(a, b ∈ T) is defined by
We shall often note fΔ(t) by
$\begin{array}{}
\frac{\Delta}{\Delta t}
\end{array}$f(t) if f is a composition of other functions. Furthermore, if f and g are both differentiable, the next formulate hold
Lagrange equations for the relative motion systems on time scales
Equation of Chetaev constraint the relative motion systems
We know that the motion of a complex system may include the motion of a carrier, as well as the motion of a carried system relative to the carrier.
Suppose that the velocity of the base point in a carrier v0 and its angular velocity is ω. We assume that the motion of N particles wouldn’t change the motion rule of the carrier which is predetermined. N generalized coordinates qs (s = 1, ⋯, n) determine the configuration of systems. If the movement of the systems are constrained by the double-sided ideal Chetaev nonholonomic constraints,
where T is the kinetic energy of the relative motion function, λβ is Lagrange multiplier, Qs, V0, Vω,
$\begin{array}{}
Q^{\omega}_{s}
\end{array}$
, Γs are respectively the generalized forces, the potential energy of uniform force field, the potential energy of inertial centrifugal force field, generalized rotary inertia force, generalized gyroscopic force.
The Qs can be divided into parts of potential and nonpotential
The action S is said to be quasi invariant on U under the transformation groups (23), if and only if for any subinterval [ta, tb] ∈ [a, b], any ε, q ∈ U
where the transformations satisfy the condition Eq. (26). we say that the invariance is called Noether generalized quasi-symmetry of the relative motion systems on time scales.
Theorem 1
If the action S is quasi-invariant on the infinitesimal transformations Eq. (23) then
For Chetaev constraint the relative motion systems on time scales, if the infinitesimal transformations Eq. (23) satisfy the conditions Eq. (26), then the system Eq. (22) has conserved quantities of the form
Where ξ0, ξs : [a, b] × Rn → R are delta differentiable functions.
In this case, we assume the map t ∈ [a, b] ↦ α (t) = t∗ ∈ R is a strictly increasing
$\begin{array}{}
C^{1}_{rd}
\end{array}$
function and its image is a new time scale t∗ = α(t), whose forward jump operator and delta derivative are denote by σ∗ and Δ∗. Following the arguments provided above,
Noether theory points out that ξ0, ξs satisfy the Noether identity if it exists the gauge G = G(t,
$\begin{array}{}
q^{\sigma}_{s}, q^{\Delta}_{s}
\end{array}$
). If the action S is generalized quasi-invariant in the infinitesimal transformations Eq. (30), then
All of the above equation, we obtain the Noether identity Eq. (33) of the relative motion systems with Chetaev type constraints on time scales.
Theorem 4
For Chetaev constraint the relative motion systems on time scales, if the infinitesimal transformations Eq. (30) satisfy the conditions Eq. (32), then the system Eq. (22) has conserved quantities of the form
According to the above proof, we can learn that Eq. (39) is called the Noether’ s conserved quantities for the relative motion systems with Chetaev type constraints on time scales.
Examples
We first consider an example of the relative motion systems, the time scale is:
In this paper, based on the theory of calculus on time scale and variational principle, we studied Noether symmetries and the conservation laws of relative motion systems on time scales. The results have shown significant approaches to seek conservation laws for these systems and provide a good method for solving the practical problems such asbiology, thermodynamics, engineering and so on.