Noether’s theorems for the relative motion systems on time scales

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

$\begin{matrix} σ (t) = i n f {s \in T : s > t}, \end{matrix}$ $$\begin{array}{} \displaystyle \sigma(t)=inf\{s\in T:s \gt t\}, \end{array}$$(1)

and the backward jump operator ρ : T → T by

$\begin{matrix} ρ (t) = s u p {s \in T : s < t} . \end{matrix}$ $$\begin{array}{} \displaystyle \rho(t)=sup\{s\in T:s \lt t\}. \end{array}$$(2)

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 T^k = T − {M} if T has a left-scattered maximum M, otherwise T^k = T.

Definition 2

Assume f : T → R is a function and t ∈ T^k, we define f^Δ(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

$\begin{matrix} | [f (σ (t)) - f (s)] - f^{Δ} (t) [σ (t) - s] | \leq ε | σ (t) - s |, \end{matrix}$ $$\begin{array}{} \displaystyle |[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]|\leq\varepsilon|\sigma(t)-s|, \end{array}$$(3)

is true for all s ∈ U, we call f^Δ(t) the delta derivative of f at t.

Definition 3

A function f : T → R is continuous at t ∈ T^k and t is right-scattered, then f is differentiable at t with

$\begin{matrix} f^{Δ} (t) = \frac{f (σ (t)) - f (t)}{μ (t)} . \end{matrix}$ $$\begin{array}{} \displaystyle f^{\Delta}(t)=\frac{f\left(\sigma(t)\right)-f(t)}{\mu(t)}. \end{array}$$(4)

Furthermore, if f is differentiable at t ∈ T^k, then

$\begin{matrix} f (σ (t)) = f (t) + μ (t) f^{Δ} (t) . \end{matrix}$ $$\begin{array}{} \displaystyle f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t). \end{array}$$(5)

Definition 4

Assume f : T → R is a regulated function, existing a function F with F^Δ(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

$\begin{matrix} \int_{a}^{b} f (t) Δ t = F (b) - F (a), \end{matrix}$ $$\begin{array}{} \displaystyle \int^{b}_{a}f(t)\Delta t=F(b)-F(a), \end{array}$$(6)

We define the indefinite integral of f by

$\begin{matrix} \int f (t) Δ t = F (t) + C . \end{matrix}$ $$\begin{array}{} \displaystyle \int f(t)\Delta t=F(t)+C. \end{array}$$(7)

Where C is an arbitrary constant.

We shall often note f^Δ(t) by $\begin{matrix} \frac{Δ}{Δ t} \end{matrix}$ $\begin{array}{} \frac{\Delta}{\Delta t} \end{array}$f(t) if f is a composition of other functions. Furthermore, if f and g are both differentiable, the next formulate hold

$\begin{matrix} (f + g)^{Δ} (t) = f^{Δ} (t) + g^{Δ} (t), \\ (k f)^{Δ} (t) = k f^{Δ} (t), \\ (f g)^{Δ} (t) = f^{Δ} (t) g (t) + f^{σ} (t) g^{Δ} (t) + f^{Δ} (t) g^{σ} (t), \end{matrix}$ $$\begin{array}{} \displaystyle (f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t),\\ \displaystyle(kf)^{\Delta}(t)=kf^{\Delta}(t),\\ \displaystyle(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f^{\sigma}(t)g^{\Delta}(t)+f^{\Delta}(t)g^{\sigma}(t), \end{array}$$(8)

where we abbreviate f ∘ σ = f(σ(t)) by f^σ.

Remark 1

we consider the two cases T = R and T = Z.

(i)

If T = R, f : R → R is delta differentiable at t ∈ R, then f^Δ(t) = f′(t).

(ii)

If T = Z, f : Z → R is delta derivative at every t ∈ Z with f^Δ(t) = f(t + 1) − f(t).

Lemma 1

(Dubois-Reymond) [6] Let g ∈ C_rd, g : [a, b] → Rⁿ, if

$\begin{matrix} \int_{a}^{b} g^{T} η^{Δ} (t) Δ t = 0, \end{matrix}$ $$\begin{array}{} \displaystyle \int^{b}_{a}g^{T}\eta^{\Delta}(t)\Delta t=0, \end{array}$$(9)

for all η ∈ $\begin{matrix} C_{r d}^{1} \end{matrix}$ $\begin{array}{} C^{1}_{rd} \end{array}$ with η(a) = η(b) = 0, then,

$\begin{matrix} g (t) \equiv c, \forall t \in [a, b]^{k} f o r s o m e c \in R^{n}, \end{matrix}$ $$\begin{array}{} \displaystyle g(t)\equiv c, \forall t \in[a,b]^{k} for some c\in R^{n}, \end{array}$$

where $\begin{matrix} C_{r d}^{1} \end{matrix}$ $\begin{array}{} C^{1}_{rd} \end{array}$ means the set of differentiable functions with rd-continuous derivative.

We can also obtain the following conclusions about the time scales [25].

Assume that α : T → R is strictly increasing and T^∗: = α(T) is a time scale, Let β : T^∗ → R, then there exists t in the real interval [t, σ(t)] with

$\begin{matrix} (α \circ β)^{Δ} (t) = β^{Δ^{*}} (α (t)) α^{Δ} (t), \\ \frac{1}{α^{Δ}} = (α^{- 1})^{Δ^{*}} \circ α . \end{matrix}$ $$\begin{array}{} \displaystyle (\alpha\circ\beta)^{\Delta}(t)=\beta^{\Delta^{*}}(\alpha(t))\alpha^{\Delta}(t),\\ \displaystyle\frac{1}{\alpha^{\Delta}}=(\alpha^{-1})^{\Delta^{*}}\circ\alpha. \end{array}$$(10)

If f : T → R is an rd-continuous function and α is differentiable with rd-continuous derivative, then for a, b ∈ T,

$\begin{matrix} \int_{a}^{^{b}} f (t) α^{Δ} (t) Δ t = \int_{α (a)}^{α (b)} (f \circ α^{- 1}) (t) Δ^{*} t = \int_{α (a)}^{α (b)} f (t^{*}) Δ^{*} t . \end{matrix}$ $$\begin{array}{} \displaystyle \int^{^{b}}_{a}f(t)\alpha^{\Delta}(t)\Delta t=\int^{\alpha(b)}_{\alpha(a)}\left(f\circ\alpha^{-1}\right)(t)\Delta^{*}t=\int^{\alpha(b)}_{\alpha(a)}f(t^{*})\Delta^{*}t. \end{array}$$(11)

Let γ = α⁻¹, then q^∗(t^∗) := Q_ε(γ(t^∗), q(γ(t^∗))).

Lagrange equations for the relative motion systems on time scales

3.1

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 q_s (s = 1, ⋯, n) determine the configuration of systems. If the movement of the systems are constrained by the double-sided ideal Chetaev nonholonomic constraints,

$\begin{matrix} f_{β} (q_{s}, \dot{q_{s}}, t) = 0 (β = 1, \dots, g; s = 1, \dots, n) . \end{matrix}$ $$\begin{array}{} \displaystyle f_{\beta}(q_{s},\dot{q_{s}},t)=0(\beta=1,\cdots,g;s=1,\cdots,n). \end{array}$$

The equations of the relative motion systems are[18]

$\begin{matrix} (\begin{matrix} \frac{d}{d t} \frac{\partial T}{\partial \dot{q_{s}}} - \frac{\partial T}{\partial q_{s}} = Q_{s} - \frac{\partial}{\partial q_{s}} (V^{0} + V^{ω}) + Q_{s}^{ω} + Γ_{s} + Λ_{s}, \\ Λ_{s} = λ_{β} \frac{\partial f_{β}}{\partial \dot{q_{s}}} (s = 1, \dots, n), \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left \{\begin{array}{} \quad\frac{d}{dt}\frac{\partial T}{\partial \dot{q_{s}}}-\frac{\partial T}{\partial q_{s}}=Q_{s}-\frac{\partial}{\partial q_{s}}(V^{0}+V^{\omega})+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s},\\ \\ \quad\Lambda_{s}=\lambda_{\beta}\frac{\partial f_{\beta}}{\partial \dot{q_{s}}} (s=1,\cdots,n), \end{array} \right. \end{array}$$

where T is the kinetic energy of the relative motion function, λ_β is Lagrange multiplier, Q_s, V⁰, V^ω, $\begin{matrix} Q_{s}^{ω} \end{matrix}$ $\begin{array}{} Q^{\omega}_{s} \end{array}$ , Γ_s 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 Q_s can be divided into parts of potential and nonpotential

$\begin{matrix} Q_{s} = Q_{s}^{'} + Q_{s}^{″}, Q_{s}^{'} = \frac{\partial V}{\partial q_{s}} \end{matrix}$ $$\begin{array}{} \displaystyle Q_{s}=Q'_{s}+Q''_{s},Q'_{s}=\frac{\partial V}{\partial q_{s}} \end{array}$$

We construct Lagrangian of the relative motion system

$\begin{matrix} L = T - V - V^{0} - V^{ω} . \end{matrix}$ $$\begin{array}{} \displaystyle L=T-V-V^{0}-V^{\omega}. \end{array}$$

Equations can be written as follows

$\begin{matrix} \frac{d}{d t} \frac{\partial L}{\partial \dot{q_{s}}} - \frac{\partial L}{\partial q_{s}} = Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{q_{s}}}-\frac{\partial L}{\partial q_{s}}=Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s}. \end{array}$$

3.2

Lagrange equation for the relative motion systems with delta derivatives

Firstly, we introduce the following relationships [13]

$\begin{matrix} \frac{Δ}{Δ t} (δ q_{s}) = δ (\frac{Δ}{Δ t} q_{s}) = δ q_{s}^{Δ}, \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\Delta}{\Delta t}(\delta q_{s})=\delta\left(\frac{\Delta}{\Delta t}q_{s}\right)=\delta q^{\Delta}_{s}, \end{array}$$(12)

$\begin{matrix} (δ q_{s})^{σ} = δ q_{s}^{σ} . \end{matrix}$ $$\begin{array}{} \displaystyle (\delta q_{s})^{\sigma}=\delta q^{\sigma}_{s}. \end{array}$$(13)

The Hamilton principle for the relative motion system on time scales is written by

$\begin{matrix} \int_{t_{a}}^{t_{b}} [δ L + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s}) δ q_{s}^{σ}] Δ t = 0, \end{matrix}$ $$\begin{array}{} \displaystyle \int^{t_{b}}_{t_{a}}[\delta L+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s})\delta q^{\sigma}_{s}]\Delta t=0, \end{array}$$(14)

where $\begin{matrix} (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s}) δ q_{s}^{σ} \end{matrix}$ $\begin{array}{} (Q''_{s}+Q^{\omega}_{s}+\Gamma_{s})\delta q^{\sigma}_{s} \end{array}$ is the virtual work of generalized force.

We take total variation for Lagrange function L, then

$\begin{matrix} δ L = \frac{\partial L}{\partial q_{s}^{σ}} δ q_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} δ q_{s}^{Δ} . \end{matrix}$ $$\begin{array}{} \displaystyle \delta L=\frac{\partial L}{\partial q^{\sigma}_{s}}\delta q^{\sigma}_{s}+\frac{\partial L}{\partial q^{\Delta}_{s}}\delta q^{\Delta}_{s}. \end{array}$$(15)

By using of Eq. (15) and Eq. (14), we can obtain

$\begin{matrix} \int_{t_{a}}^{t_{b}} (\frac{\partial L}{\partial q_{s}^{σ}} δ q_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} δ q_{s}^{Δ} + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s}) δ q_{s}^{σ}) Δ t \\ = \int_{t_{a}}^{t_{b}} ((\frac{\partial L}{\partial q_{s}^{σ}} + Q_{s}^{″} + Q_{s}^{ω} + Γ_{s}) δ q_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} δ q_{s}^{Δ}) Δ t \\ = \int_{t_{a}}^{t_{b}} ((\frac{\partial L}{\partial q_{s}^{σ}} + Q_{s}^{″} + Q_{s}^{ω} + Γ_{s}) (δ q_{s})^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} (δ q_{s})^{Δ}) Δ t \\ = \int_{t_{a}}^{t_{b}} {(\int_{t_{a}}^{t} (\frac{Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + \partial L (τ, q_{s}^{σ} (τ), q_{s}^{Δ} (τ))}{\partial q_{s}^{σ} (τ)}) Δ τ \cdot δ q_{s})}^{Δ} Δ t \\ - \int_{t_{a}}^{t_{b}} (\int_{t_{a}}^{t} (\frac{Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + \partial L (τ, q_{s}^{σ} (τ), q_{s}^{Δ} (τ))}{\partial q_{s}^{σ} (τ)}) Δ τ) (δ q_{s})^{Δ} Δ t + \int_{t_{a}}^{t_{b}} \frac{\partial L}{\partial q_{s}^{Δ}} (δ q_{s})^{Δ} Δ t \\ = \int_{t_{a}}^{t_{b}} (\frac{\partial L}{\partial q_{s}^{Δ}} - \int_{t_{a}}^{t} (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + \frac{\partial L (τ, q_{s}^{σ} (τ), q_{s}^{Δ} (τ))}{\partial q_{s}^{σ} (τ)}) Δ τ) (δ q_{s})^{Δ} Δ t \\ = 0 \end{matrix}$ $$\begin{array}{} \displaystyle \int^{t_{b}}_{t_{a}}\left[\frac{\partial L}{\partial q^{\sigma}_{s}}\delta q^{\sigma}_{s}+\frac{\partial L}{\partial q^{\Delta}_{s}}\delta q^{\Delta}_{s}+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s})\delta q^{\sigma}_{s}\right]\Delta t \\ \displaystyle=\int^{t_{b}}_{t_{a}}\left[\left(\frac{\partial L}{\partial q^{\sigma}_{s}}+Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}\right)\delta q^{\sigma}_{s}+\frac{\partial L}{\partial q^{\Delta}_{s}}\delta q^{\Delta}_{s}\right]\Delta t \\ \displaystyle=\int^{t_{b}}_{t_{a}}\left[\left(\frac{\partial L}{\partial q^{\sigma}_{s}}+Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}\right)(\delta q_{s})^{\sigma}+\frac{\partial L}{\partial q^{\Delta}_{s}}(\delta q_{s})^{\Delta}\right]\Delta t \\ \displaystyle=\int^{t_{b}}_{t_{a}}\left[\int^{t}_{t_{a}}\left(\frac{Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\partial L(\tau,q^{\sigma}_{s}(\tau),q^{\Delta}_{s}(\tau))}{\partial q^{\sigma}_{s}(\tau)}\right)\Delta \tau\cdot\delta q_{s}\right]^{\Delta}\Delta t\\ \displaystyle-\int^{t_{b}}_{t_{a}}\left[\int^{t}_{t_{a}}\left(\frac{Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\partial L(\tau,q^{\sigma}_{s}(\tau),q^{\Delta}_{s}(\tau))}{\partial q^{\sigma}_{s}(\tau)}\right)\Delta \tau\right](\delta q_{s})^{\Delta}\Delta t+\int^{t_{b}}_{t_{a}}\frac{\partial L}{\partial q^{\Delta}_{s}}(\delta q_{s})^{\Delta}\Delta t\\ \displaystyle\displaystyle=\int^{t_{b}}_{t_{a}}\left[\frac{\partial L}{\partial q^{\Delta}_{s}}-\int^{t}_{t_{a}}\left(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\frac{\partial L(\tau,q^{\sigma}_{s}(\tau),q^{\Delta}_{s}(\tau))}{\partial q^{\sigma}_{s}(\tau)}\right)\Delta\tau\right](\delta q_{s})^{\Delta}\Delta t\\ \displaystyle=0 \end{array}$$

Therefore, by Lemma 1

$\begin{matrix} \frac{\partial L}{\partial q_{s}^{Δ}} - \int_{t_{a}}^{t} (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + \frac{\partial L (τ, q_{s}^{σ} (τ), q_{s}^{Δ} (τ))}{\partial q_{s}^{σ} (τ)}) Δ τ \equiv c o n s t, t \in [t_{a}, t_{b}] . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial L}{\partial q^{\Delta}_{s}}-\int^{t}_{t_{a}}\left(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\frac{\partial L(\tau,q^{\sigma}_{s}(\tau),q^{\Delta}_{s}(\tau))}{\partial q^{\sigma}_{s}(\tau)}\right)\Delta\tau\equiv const,t\in[t_{a},t_{b}]. \end{array}$$

Hence

$\begin{matrix} \frac{Δ}{Δ t} \frac{\partial L}{\partial q_{s}^{Δ}} - \frac{\partial L}{\partial q_{s}^{σ}} - (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s}) = 0 \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\Delta}{\Delta t}\frac{\partial L}{\partial q^{\Delta}_{s}}-\frac{\partial L}{\partial q^{\sigma}_{s}}-(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s})=0 \end{array}$$(16)

Assuming the movement of the system is constrained by the double-sided ideal nonholonomic of Chetaev type with delta derivatives

$\begin{matrix} f_{β} (t, q_{s}^{σ}, q_{s}^{Δ}) = 0, (β = 1, 2, \dots, g) . \end{matrix}$ $$\begin{array}{} \displaystyle f_{\beta}(t,q^{\sigma}_{s},q^{\Delta}_{s})=0,(\beta=1,2,\cdots,g). \end{array}$$(17)

Suppose the restrictions that constraints impose on the virtual displacements are

$\begin{matrix} \frac{\partial f_{β}}{\partial q_{s}^{Δ}} δ q_{s}^{σ} = 0, (s = 1, 2, \dots, n; β = 1, 2, \dots, g) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial f_{\beta}}{\partial q^{\Delta}_{s}}\delta q^{\sigma}_{s}=0,(s=1,2,\cdots,n;\beta=1,2,\cdots,g). \end{array}$$(18)

Multiplying $\begin{matrix} δ q_{s}^{σ} \end{matrix}$ $\begin{array}{} \delta q^{\sigma}_{s} \end{array}$ on both sides of Eq. (16)

$\begin{matrix} ((Q_{s}^{″} + Q_{s}^{ω} + Γ_{s}) - \frac{Δ}{Δ t} \frac{\partial L}{\partial q_{s}^{Δ}} + \frac{\partial L}{\partial q_{s}^{σ}}) \cdot δ q_{s}^{σ} = 0. \end{matrix}$ $$\begin{array}{} \displaystyle \left((Q''_{s}+Q^{\omega}_{s}+\Gamma_{s})-\frac{\Delta}{\Delta t}\frac{\partial L}{\partial q^{\Delta}_{s}}+\frac{\partial L}{\partial q^{\sigma}_{s}}\right)\cdot\delta q^{\sigma}_{s}=0. \end{array}$$(19)

Introducing the Lagrange multiplier λ and multiplying λ on both sides of Eq. (18),

$\begin{matrix} λ \frac{\partial f_{β}}{\partial q_{s}^{Δ}} δ q_{s}^{σ} = 0. \end{matrix}$ $$\begin{array}{} \displaystyle \lambda\frac{\partial f_{\beta}}{\partial q^{\Delta}_{s}}\delta q^{\sigma}_{s}=0. \end{array}$$(20)

Form Eq. (20) and Eq. (19), we get

$\begin{matrix} ((Q_{s}^{″} + Q_{s}^{ω} + Γ_{s}) - \frac{Δ}{Δ t} \frac{\partial L}{\partial q_{s}^{Δ}} + \frac{\partial L}{\partial q_{s}^{σ}} + λ \frac{\partial f_{β}}{\partial q_{s}^{Δ}}) \cdot δ q_{s}^{σ} = 0. \end{matrix}$ $$\begin{array}{} \displaystyle \left((Q''_{s}+Q^{\omega}_{s}+\Gamma_{s})-\frac{\Delta}{\Delta t}\frac{\partial L}{\partial q^{\Delta}_{s}}+\frac{\partial L}{\partial q^{\sigma}_{s}}+\lambda\frac{\partial f_{\beta}}{\partial q^{\Delta}_{s}}\right)\cdot\delta q^{\sigma}_{s}=0. \end{array}$$(21)

Differential Eq. (21), we obtain the equation of the relative motion systems with Chetaev type constraints on time scales

$\begin{matrix} \frac{Δ}{Δ t} \frac{\partial L}{\partial q_{s}^{Δ}} - \frac{\partial L}{\partial q_{s}^{σ}} = Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s} . \\ (Λ_{s} = λ \frac{\partial f_{β}}{\partial q_{s}^{Δ}}) \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\Delta}{\Delta t}\frac{\partial L}{\partial q^{\Delta}_{s}}-\frac{\partial L}{\partial q^{\sigma}_{s}}=Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s}.\\ \displaystyle(\Lambda_{s}=\lambda\frac{\partial f_{\beta}}{\partial q^{\Delta}_{s}}) \end{array}$$(22)

Noether theorem of the relative motion systems

4.1

Noether’s theorem without transforming time

The Hamilton action with the delta derivative on time scales can be expressed as

$\begin{matrix} S (γ) = \int_{t_{a}}^{t_{b}} L (t, q_{s}^{σ}, q_{s}^{Δ}) Δ t, \end{matrix}$ $$\begin{array}{} \displaystyle S(\gamma)=\int^{t_{b}}_{t_{a}}L(t,q^{\sigma}_{s},q^{\Delta}_{s})\Delta t, \end{array}$$

where γ is a curve.

Introducing the following single parameter infinitesimal transformations without transforming time:

$\begin{matrix} t^{*} = t, \\ q_{s}^{*} = q_{s} + ε ξ_{s} (t, q) + o (ε), \end{matrix}$ $$\begin{array}{} \displaystyle t^{*}=t, \\ \displaystyle q^{*}_{s}=q_{s}+\varepsilon\xi_{s}(t,q)+o(\varepsilon), \end{array}$$(23)

if and only if

$\begin{matrix} \int_{t_{a}}^{t_{b}} L (t, q_{s}^{σ}, q_{s}^{Δ}) Δ t = \int_{t_{a}}^{t_{b}} L (t, q_{s}^{* σ}, q_{s}^{* Δ}) Δ t . \end{matrix}$ $$\begin{array}{} \displaystyle \int^{t_{b}}_{t_{a}}L(t,q^{\sigma}_{s},q^{\Delta}_{s})\Delta t=\int^{t_{b}}_{t_{a}}L(t,q^{*\sigma}_{s},q^{*\Delta}_{s})\Delta t. \end{array}$$(24)

Where ε is the infinitesimal parameter, ξ_s : [a, b] × Rⁿ → R is delta differentiable functions.

The relationship between the isochronous δ and the total variation Δ on time scale is as follows

$\begin{matrix} Δ q_{s}^{σ} = δ q_{s}^{σ} + q_{s}^{Δ} Δ t . \end{matrix}$ $$\begin{array}{} \displaystyle \Delta q^{\sigma}_{s}=\delta q^{\sigma}_{s}+q^{\Delta}_{s}\Delta t. \end{array}$$

According to Eq. (23), we have

$\begin{matrix} δ q_{s}^{σ} = Δ q_{s}^{σ} - q_{s}^{Δ} Δ t = ε ξ_{s}^{σ} . \end{matrix}$ $$\begin{array}{} \displaystyle \delta q^{\sigma}_{s}=\Delta q^{\sigma}_{s}-q^{\Delta}_{s}\Delta t=\varepsilon\xi^{\sigma}_{s}. \end{array}$$(25)

Substituting Eq. (25) into Eq. (18) has

$\begin{matrix} \frac{\partial f_{β}}{\partial q_{s}^{Δ}} ξ_{s}^{σ} = 0, (s = 1, 2, \dots, n; β = 1, 2, \dots, g) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial f_{\beta}}{\partial q^{\Delta}_{s}}\xi^{\sigma}_{s}=0,(s=1,2,\cdots,n;\beta=1,2,\cdots,g). \end{array}$$(26)

Definition 5

The action S is said to be quasi invariant on U under the transformation groups (23), if and only if for any subinterval [t_a, t_b] ∈ [a, b], any ε, q ∈ U

$\begin{matrix} \int_{t_{a}}^{t_{b}} L (t, q_{s}^{σ}, q_{s}^{Δ}) Δ t = \int_{t_{a}}^{t_{b}} L (t, q_{s}^{* σ}, q_{s}^{* Δ}) Δ t + \int_{t_{a}}^{t_{b}} (\frac{Δ}{Δ t} (Δ G) + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) δ q_{s}^{σ}) Δ t, \end{matrix}$ $$\begin{array}{} \displaystyle \int^{t_{b}}_{t_{a}}L(t,q^{\sigma}_{s},q^{\Delta}_{s})\Delta t=\int^{t_{b}}_{t_{a}}L(t,q^{*\sigma}_{s},q^{*\Delta}_{s})\Delta t +\int^{t_{b}}_{t_{a}}\left(\frac{\Delta}{\Delta t}(\Delta G)+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\delta q^{\sigma}_{s}\right)\Delta t, \end{array}$$

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

$\begin{matrix} \frac{\partial L}{\partial q_{s}^{σ}} ξ_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) ξ_{s}^{σ} = - \frac{Δ}{Δ t} G, \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial L}{\partial q^{\sigma}_{s}}\xi^{\sigma}_{s}+\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\xi^{\sigma}_{s}=-\frac{\Delta}{\Delta t}G, \end{array}$$(27)

where $\begin{matrix} ξ_{s}^{σ} (t, q) = ξ_{s} (σ (t), q (σ (t))), ξ_{s}^{Δ} (t, q) = \frac{Δ}{Δ t} ξ_{s} (t, q) \end{matrix}$ $\begin{array}{} \xi^{\sigma}_{s}(t,q)=\xi_{s}(\sigma(t), q(\sigma(t))),\,\, \xi^{\Delta}_{s}(t,q)=\frac{\Delta}{\Delta t}\xi_{s}(t,q) \end{array}$.

Proof

Consider the infinitesimal transformations (t, $\begin{matrix} q_{s}^{*} \end{matrix}$ $\begin{array}{} q^{*}_{s} \end{array}$ ) given by group (23) and Definition 5, we can obtain

$\begin{matrix} \int_{t_{a}}^{t_{b}} L (t, q_{s}^{σ}, q_{s}^{Δ}) Δ t = \int_{t_{a}}^{t_{b}} L (t, q_{s}^{* σ}, q_{s}^{* Δ}) Δ t \\ + \int_{t_{a}}^{t_{b}} (\frac{Δ}{Δ t} (Δ G) + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) δ q_{s}^{σ}) Δ t \\ = \int_{t_{a}}^{t_{b}} L (t, q_{s}^{σ} + ε ξ_{s}^{σ} + o (ε), q_{s}^{Δ} + ε ξ_{s}^{Δ} + o (ε)) Δ t \\ + \int_{t_{a}}^{t_{b}} (\frac{Δ}{Δ t} (Δ G) + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) δ q_{s}^{σ}) Δ t \end{matrix}$ $$\begin{array}{} \displaystyle \int^{t_{b}}_{t_{a}}L(t,q^{\sigma}_{s},q^{\Delta}_{s})\Delta t=\int^{t_{b}}_{t_{a}}L(t,q^{*\sigma}_{s},q^{*\Delta}_{s})\Delta t \\ \displaystyle\qquad\qquad\qquad\qquad+\int^{t_{b}}_{t_{a}}\left(\frac{\Delta}{\Delta t}(\Delta G)+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\delta q^{\sigma}_{s}\right)\Delta t\\ \displaystyle\qquad\qquad\qquad\qquad=\int^{t_{b}}_{t_{a}}L(t,q^{\sigma}_{s}+\varepsilon\xi^{\sigma}_{s}+o(\varepsilon),q^{\Delta}_{s}+\varepsilon\xi^{\Delta}_{s}+o(\varepsilon))\Delta t \\ \displaystyle\qquad\qquad\qquad\qquad+\int^{t_{b}}_{t_{a}}\left(\frac{\Delta}{\Delta t}(\Delta G)+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\delta q^{\sigma}_{s}\right)\Delta t \end{array}$$

Taking into account any subinterval [t_a, t_b] ∈ [a, b], we can get the following equivalent equation:

$\begin{matrix} L (t, q_{s}^{σ}, q_{s}^{Δ}) = L (t, q_{s}^{σ} + ε ξ_{s}^{σ} + o (ε), q_{s}^{Δ} + ε ξ_{s}^{Δ} + o (ε)) + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) Δ q_{s}^{σ} + \frac{Δ}{Δ t} (Δ G) . \end{matrix}$ $$\begin{array}{} \displaystyle L(t,q^{\sigma}_{s},q^{\Delta}_{s})=L(t,q^{\sigma}_{s}+\varepsilon\xi^{\sigma}_{s}+o(\varepsilon),q^{\Delta}_{s}+\varepsilon\xi^{\Delta}_{s}+o(\varepsilon)) +(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\Delta q^{\sigma}_{s}+\frac{\Delta}{\Delta t}(\Delta G). \end{array}$$(28)

Differentiation both sides of Eq. (28) with respect to ε,

$\begin{matrix} \frac{\partial L}{\partial t} \frac{\partial t}{\partial ε} + \frac{\partial L}{\partial q_{s}^{σ}} \frac{\partial q_{s}^{* σ}}{\partial ε} + \frac{\partial L}{\partial q_{s}^{Δ}} \frac{\partial q_{s}^{* Δ}}{\partial ε} + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) \frac{\partial δ q_{s}^{σ}}{\partial ε} + \frac{Δ}{Δ t} G = 0 \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial L}{\partial t}\frac{\partial t}{\partial \varepsilon}+\frac{\partial L}{\partial q^{\sigma}_{s}}\frac{\partial q^{*\sigma}_{s}}{\partial \varepsilon}+\frac{\partial L}{\partial q^{\Delta}_{s}}\frac{\partial q^{*\Delta}_{s}}{\partial \varepsilon}+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\frac{\partial \delta q^{\sigma}_{s}}{\partial \varepsilon}+\frac{\Delta}{\Delta t}G=0 \end{array}$$

Since

$\begin{matrix} \frac{\partial q_{s}^{* σ}}{\partial ε} |_{ε = 0} = ξ_{s}^{σ}, \frac{\partial q_{s}^{* Δ}}{\partial ε} |_{ε = 0} = ξ_{s}^{Δ} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial q^{*\sigma}_{s}}{\partial \varepsilon}|_{\varepsilon=0}=\xi^{\sigma}_{s},\frac{\partial q^{*\Delta}_{s}}{\partial \varepsilon}|_{\varepsilon=0}=\xi^{\Delta}_{s}. \end{array}$$

Eq. (25) shows that

$\begin{matrix} \frac{\partial δ q_{s}^{σ}}{\partial ε} = ξ_{s}^{σ} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial \delta q^{\sigma}_{s}}{\partial \varepsilon}=\xi^{\sigma}_{s}. \end{array}$$

Then, setting ε = 0, we can obtain the Eq. (27).

Theorem 2

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

$\begin{matrix} I = \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s} + G . \end{matrix}$ $$\begin{array}{} \displaystyle I=\frac{\partial L}{\partial q^{\Delta}_{s}}\xi_{s}+G. \end{array}$$(29)

Proof

It proves that Eq. (29) is equivalent to the proof of I = const, takeing the derivative of I with respect to t, then

$\begin{matrix} \frac{Δ}{Δ t} I = \frac{Δ}{Δ t} (\frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s} + G) = \frac{Δ}{Δ t} \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} + \frac{Δ}{Δ t} G . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\Delta}{\Delta t}I=\frac{\Delta}{\Delta t}\left[\frac{\partial L}{\partial q^{\Delta}_{s}}\xi_{s}+G\right]=\frac{\Delta}{\Delta t}\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\sigma}_{s}+\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}+\frac{\Delta}{\Delta t}G. \end{array}$$

Multiplying $\begin{matrix} ξ_{s}^{σ} \end{matrix}$ $\begin{array}{} \xi^{\sigma}_{s} \end{array}$ on both sides of Eq.(22)

$\begin{matrix} \frac{Δ}{Δ t} \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{σ} = \frac{\partial L}{\partial q_{s}^{σ}} ξ_{s}^{σ} + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) ξ_{s}^{σ} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\Delta}{\Delta t}\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\sigma}_{s}=\frac{\partial L}{\partial q^{\sigma}_{s}}\xi^{\sigma}_{s}+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\xi^{\sigma}_{s}. \end{array}$$

Observe that Eq. (27)

$\begin{matrix} \frac{Δ}{Δ t} I = \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s} + \frac{\partial L}{\partial q_{s}^{σ}}) ξ_{s}^{σ} + \frac{Δ}{Δ t} G = 0. \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\Delta}{\Delta t}I=\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}+\left(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s}+\frac{\partial L}{\partial q^{\sigma}_{s}}\right)\xi^{\sigma}_{s}+\frac{\Delta}{\Delta t}G=0. \end{array}$$

Namely

$\begin{matrix} I = c o n s t . \end{matrix}$ $$\begin{array}{} \displaystyle I=const. \end{array}$$

4.2

Noether theorem with transforming time

Considering the following infinitesimal transformations with the time and the state variables:

$\begin{matrix} t^{*} = t + ε ξ_{0} (t, q) + o (ε), \\ q_{s}^{*} = q_{s} + ε ξ_{s} (t, q) + o (ε) . \end{matrix}$ $$\begin{array}{} \displaystyle t^{*}=t+\varepsilon\xi_{0}(t,q)+o(\varepsilon),\\ \displaystyle q^{*}_{s}=q_{s}+\varepsilon\xi_{s}(t,q)+o(\varepsilon). \end{array}$$(30)

Where ξ₀, ξ_s : [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 $\begin{matrix} C_{r d}^{1} \end{matrix}$ $\begin{array}{} C^{1}_{rd} \end{array}$ 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,

$\begin{matrix} σ^{*} \circ α = α \circ σ . \end{matrix}$ $$\begin{array}{} \displaystyle \sigma^{*}\circ\alpha=\alpha\circ\sigma. \end{array}$$

According to groups (30), we have

$\begin{matrix} δ q_{s}^{σ} = Δ q_{s}^{σ} - q_{s}^{Δ} Δ t = ε (ξ_{s}^{σ} - q_{s}^{σ Δ} ξ_{0}^{σ}) . \end{matrix}$ $$\begin{array}{} \displaystyle \delta q^{\sigma}_{s}=\Delta q^{\sigma}_{s}-q^{\Delta}_{s}\Delta t=\varepsilon(\xi^{\sigma}_{s}-q^{\sigma\Delta}_{s}\xi^{\sigma}_{0}). \end{array}$$(31)

Substituting Eq. (31) into Eq. (18)

$\begin{matrix} f_{β s} (ξ_{s}^{σ} - q_{s}^{σ Δ} ξ_{0}^{σ}) = 0 \end{matrix}$ $$\begin{array}{} \displaystyle f_{\beta s}(\xi^{\sigma}_{s}-q^{\sigma\Delta}_{s}\xi^{\sigma}_{0})=0 \end{array}$$(32)

Definition 6

The action S is called a generalized quasi invariant in the transformation groups (30) and if and only if for any subinterval [t_a, t_b] ∈ [a, b]

$\begin{matrix} \int_{t_{a}}^{t_{b}} L (t, q_{s}^{σ}, q_{s}^{Δ}) Δ t = \int_{α (t_{a})}^{α (t_{b})} L (t^{*}, q_{s}^{* σ^{*}} (t^{*}), q_{s}^{* Δ^{*}} (t^{*})) Δ^{*} t^{*} \\ + \int_{t_{a}}^{t_{b}} (\frac{Δ}{Δ t} (Δ G) + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) δ q_{s}^{σ}) Δ t . \end{matrix}$ $$\begin{array}{} \displaystyle \int^{t_{b}}_{t_{a}}L(t,q^{\sigma}_{s},q^{\Delta}_{s})\Delta t=\int^{\alpha(t_{b})}_{\alpha(t_{a})}L\left(t^{*},q^{*\sigma^{*}}_{s}(t^{*}),q^{*\Delta^{*}}_{s}(t^{*})\right)\Delta^{*} t^{*} \\ \displaystyle\qquad\qquad\qquad\qquad\,+\int^{t_{b}}_{t_{a}}\left(\frac{\Delta}{\Delta t}(\Delta G) +(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\delta q^{\sigma}_{s}\right)\Delta t. \end{array}$$

Theorem 3

Noether theory points out that ξ₀, ξ_s satisfy the Noether identity if it exists the gauge G = G(t, $\begin{matrix} q_{s}^{σ}, q_{s}^{Δ} \end{matrix}$ $\begin{array}{} q^{\sigma}_{s}, q^{\Delta}_{s} \end{array}$ ). If the action S is generalized quasi-invariant in the infinitesimal transformations Eq. (30), then

$\begin{matrix} \frac{\partial L}{\partial t} ξ_{0} + \frac{\partial L}{\partial q_{s}^{σ}} ξ_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} + L ξ_{0}^{Δ} - \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{0}^{Δ} q_{s}^{Δ} + (Q_{s}^{″} + Q_{s}^{ω} \\ + Γ_{s} + Λ_{s}) \times [ξ_{s}^{σ} - ξ_{0}^{σ} (q_{s}^{Δ} + μ (t) q_{s}^{Δ Δ})] = - \frac{Δ}{Δ t} G . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial L}{\partial t}\xi_{0}+\frac{\partial L}{\partial q^{\sigma}_{s}}\xi^{\sigma}_{s}+\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}+L\xi^{\Delta}_{0}-\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{0}q^{\Delta}_{s}+(Q''_{s}+Q^{\omega}_{s}\\ \displaystyle+\Gamma_{s}+\Lambda_{s})\times[\xi^{\sigma}_{s}-\xi^{\sigma}_{0}(q^{\Delta}_{s}+\mu(t)q^{\Delta\Delta}_{s})]=-\frac{\Delta}{\Delta t}G. \end{array}$$(33)

Proof

Consider the Definition 6, we have

$\begin{matrix} \int_{t_{a}}^{t_{b}} L (t, q_{s}^{σ}, q_{s}^{Δ}) Δ t \\ = \int_{α (t_{a})}^{α (t_{b})} L (t^{*}, q_{s}^{* σ^{*}}, q_{s}^{* Δ^{*}}) Δ^{*} t^{*} \\ + \int_{t_{a}}^{t_{b}} (\frac{Δ}{Δ t} (Δ G) + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) δ q_{s}^{σ}) Δ t \\ = \int_{t_{a}}^{t_{b}} L (α (t), (q_{s}^{*} \circ σ^{*} \circ α) (t), q_{s}^{* Δ^{*}} (α (t))) α^{Δ} (t) Δ t \\ + \int_{t_{a}}^{t_{b}} (\frac{Δ}{Δ t} (Δ G) + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) δ q_{s}^{σ}) Δ t . \end{matrix}$ $$\begin{array}{} \displaystyle \int^{t_{b}}_{t_{a}}L(t,q^{\sigma}_{s},q^{\Delta}_{s})\Delta t\\ \displaystyle=\int^{\alpha(t_{b})}_{\alpha(t_{a})}L\left(t^{*},q^{*\sigma^{*}}_{s},q^{*\Delta^{*}}_{s}\right)\Delta^{*} t^{*}\\ \displaystyle+\int^{t_{b}}_{t_{a}}\left(\frac{\Delta}{\Delta t}(\Delta G)+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s} +\Lambda_{s})\delta q^{\sigma}_{s}\right)\Delta t\\ \displaystyle=\int^{t_{b}}_{t_{a}}L\left(\alpha(t),(q^{*}_{s}\circ\sigma^{*}\circ\alpha)(t),q^{*\Delta^{*}}_{s}(\alpha(t))\right)\alpha^{\Delta}(t)\Delta t \\ \displaystyle+\int^{t_{b}}_{t_{a}}\left(\frac{\Delta}{\Delta t}(\Delta G)+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\delta q^{\sigma}_{s}\right)\Delta t. \end{array}$$(34)

Since σ^∗ is the new forward jump operator and Δ^∗ is the new delta derivative, we can have by Eq. (10)

$\begin{matrix} (q_{s}^{*} \circ α)^{Δ} (t) = q_{s}^{* Δ^{*}} (α (t)) α^{Δ} (t) . \end{matrix}$ $$\begin{array}{} \displaystyle (q^{*}_{s}\circ\alpha)^{\Delta}(t)=q^{*\Delta^{*}}_{s}(\alpha(t))\alpha^{\Delta}(t). \end{array}$$

Then

$\begin{matrix} q_{s}^{* Δ^{*}} (α (t)) = \frac{(q_{s}^{*} \circ α)^{Δ} (t)}{α^{Δ} (t)} . \end{matrix}$ $$\begin{array}{} \displaystyle q^{*\Delta^{*}}_{s}(\alpha(t))=\frac{(q^{*}_{s}\circ\alpha)^{\Delta}(t)}{\alpha^{\Delta}(t)}. \end{array}$$(35)

Gathering Eq. (34) and Eq. (35), we can derive

$\begin{matrix} \int_{t_{a}}^{t_{b}} L (t, q_{s}^{σ}, q_{s}^{Δ}) Δ t \\ = \int_{t_{a}}^{t_{b}} L (α (t), (q_{s}^{*} \circ σ \circ α) (t), \frac{(q_{s}^{*} \circ α)^{Δ} (t)}{α^{Δ} (t)}) α^{Δ} (t) Δ t \\ + \int_{t_{a}}^{t_{b}} (\frac{Δ}{Δ t} (Δ G) + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) δ q_{s}^{σ}) Δ t . \end{matrix}$ $$\begin{array}{} \displaystyle \int^{t_{b}}_{t_{a}}L(t,q^{\sigma}_{s},q^{\Delta}_{s})\Delta t\\ \displaystyle=\int^{t_{b}}_{t_{a}}L\left(\alpha(t),(q^{*}_{s}\circ\sigma\circ\alpha)(t),\frac{(q^{*}_{s}\circ\alpha)^{\Delta}(t)}{\alpha^{\Delta}(t)}\right)\alpha^{\Delta}(t)\Delta t\\ \displaystyle+\int^{t_{b}}_{t_{a}}\left(\frac{\Delta}{\Delta t}(\Delta G)+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\delta q^{\sigma}_{s}\right)\Delta t. \end{array}$$

Since [t_a, t_b] is arbitrary subinterval

$\begin{matrix} L (t, q_{s}^{σ}, q_{s}^{Δ}) & = L (α (t), (q_{s}^{*} \circ σ \circ α) (t), \frac{(q_{s}^{*} \circ α)^{Δ} (t)}{α^{Δ} (t)}) α^{Δ} (t) \\ + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) δ q_{s}^{σ} + \frac{Δ}{Δ t} (Δ G) . \end{matrix}$ $$\begin{array}{} \displaystyle L(t,q^{\sigma}_{s},q^{\Delta}_{s})\!\!\!\!\!\!\!\!\!\!&&=L\left(\alpha(t),(q^{*}_{s}\circ\sigma\circ\alpha)(t),\frac{(q^{*}_{s}\circ\alpha)^{\Delta}(t)}{\alpha^{\Delta}(t)}\right) \alpha^{\Delta}(t)\\ &&+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\delta q^{\sigma}_{s}+\frac{\Delta}{\Delta t}(\Delta G). \end{array}$$(36)

According to group (30), differentiating both sides of Eq. (36) with respect to ε, setting ε = 0, we get

$\begin{matrix} 0 = (\frac{\partial L}{\partial t} \frac{\partial α (t)}{\partial ε} + \frac{\partial L}{\partial q_{s}^{σ}} \frac{\partial (q_{s}^{*} \circ α \circ σ)}{\partial ε} + \frac{\partial L}{\partial q_{s}^{Δ}} \frac{\partial}{\partial ε} (\frac{(q_{s}^{*} \circ α)^{Δ}}{α^{Δ}})) α^{Δ} + L \frac{\partial α^{Δ}}{\partial ε} \\ + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) \frac{\partial δ q_{s}^{σ}}{\partial ε} + \frac{Δ}{Δ t} G . \end{matrix}$ $$\begin{array}{} \displaystyle 0=\left[\frac{\partial L}{\partial t}\frac{\partial \alpha(t)}{\partial \varepsilon}+\frac{\partial L}{\partial q^{\sigma}_{s}}\frac{\partial(q^{*}_{s}\circ\alpha\circ\sigma)}{\partial \varepsilon}+\frac{\partial L}{\partial q^{\Delta}_{s}}\frac{\partial}{\partial\varepsilon}\left(\frac{(q^{*}_{s}\circ\alpha)^{\Delta}}{\alpha^{\Delta}}\right)\right]\alpha^{\Delta}+L\frac{\partial \alpha^{\Delta}}{\partial \varepsilon}\\\displaystyle\,\,\,\,+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})\frac{\partial\delta q^{\sigma}_{s}}{\partial\varepsilon}+\frac{\Delta}{\Delta t}G. \end{array}$$(37)

For the map t ∈ [a, b] ↦ α(t) ≡ t^∗ ∈ R, and $\begin{matrix} q_{s}^{*} \end{matrix}$ $\begin{array}{} q^{*}_{s} \end{array}$ ∘ α ∘ σ(t) = $\begin{matrix} q_{s}^{*} \end{matrix}$ $\begin{array}{} q^{*}_{s} \end{array}$ (α(σ(t)))

$\begin{matrix} \frac{\partial t^{*}}{\partial ε} |_{ε = 0} = ξ_{0}, \\ \frac{\partial [(q_{s}^{*} \circ α \circ σ)]}{\partial ε} |_{ε = 0} = ξ_{s}^{σ} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial t^{*}}{\partial\varepsilon}|_{\varepsilon=0}=\xi_{0},\\ \displaystyle\frac{\partial[(q^{*}_{s}\circ\alpha\circ\sigma)]}{\partial \varepsilon}|_{\varepsilon=0}=\xi^{\sigma}_{s}. \end{array}$$

For f(t) = t, then t^Δ = 1 by Remark 1, we can derive

$\begin{matrix} \frac{\partial}{\partial ε} (\frac{(q_{s}^{*} \circ α)^{Δ}}{α^{Δ}}) |_{ε = 0} & = \frac{\partial}{\partial ε} (\frac{q_{s}^{Δ} + ε ξ_{s}^{Δ} + o (ε)}{t^{Δ} + ε ξ_{0}^{Δ} + o (ε)}) \\ = \frac{ξ_{s}^{Δ} (t^{Δ} + ε ξ_{0}^{Δ}) - ξ_{0}^{Δ} (q_{s}^{Δ} + ε ξ_{s}^{Δ})}{(t^{Δ} + ε ξ_{0}^{Δ})^{2}} |_{ε = 0} \\ = ξ_{s}^{Δ} - q_{s}^{Δ} ξ_{0}^{Δ} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial}{\partial\varepsilon}\left(\frac{(q^{*}_{s}\circ\alpha)^{\Delta}}{\alpha^{\Delta}}\right)|_{\varepsilon=0}\!\!\!\!\!\!\!\!\!\! &&\displaystyle=\frac{\partial}{\partial\varepsilon}\left(\frac{q^{\Delta}_{s}+\varepsilon\xi^{\Delta}_{s}+o(\varepsilon)}{t^{\Delta}+\varepsilon\xi^{\Delta}_{0}+o(\varepsilon)}\right) \\ &&\displaystyle=\frac{\xi^{\Delta}_{s}(t^{\Delta}+\varepsilon\xi^{\Delta}_{0})-\xi^{\Delta}_{0}(q^{\Delta}_{s}+\varepsilon\xi^{\Delta}_{s})}{(t^{\Delta}+\varepsilon\xi^{\Delta}_{0})^{2}} |_{\varepsilon=0} \\ &&\displaystyle=\xi^{\Delta}_{s}-q^{\Delta}_{s}\xi^{\Delta}_{0}. \end{array}$$

and

$\begin{matrix} \frac{\partial t^{*^{Δ}}}{\partial ε} |_{ε = 0} = ξ_{0}^{Δ}, \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial t^{*^{\Delta}}}{\partial\varepsilon}|_{\varepsilon=0}=\xi^{\Delta}_{0}, \end{array}$$

$\begin{matrix} \frac{\partial δ q_{s}^{σ}}{\partial ε} |_{ε = 0} = Δ q_{s}^{σ} - q_{s}^{σ Δ} (Δ t)^{σ} |_{ε = 0} = ξ_{s}^{σ} - ξ_{0}^{σ} q_{s}^{σ Δ} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial\delta q^{\sigma}_{s}}{\partial\varepsilon}|_{\varepsilon=0}=\Delta q^{\sigma}_{s}-q^{\sigma\Delta}_{s}(\Delta t)^{\sigma}|_{\varepsilon=0}=\xi^{\sigma}_{s}-\xi^{\sigma}_{0}q^{\sigma\Delta}_{s}. \end{array}$$

Considering Definition 3, we have

$\begin{matrix} q_{s}^{σ Δ} = (q_{s}^{Δ})^{σ} = q_{s}^{Δ} + μ (t) q_{s}^{Δ Δ} . \end{matrix}$ $$\begin{array}{} \displaystyle q^{\sigma\Delta}_{s}=(q^{\Delta}_{s})^{\sigma}=q^{\Delta}_{s}+\mu(t)q^{\Delta\Delta}_{s}. \end{array}$$

Then,

$\begin{matrix} ξ_{s}^{σ} - ξ_{0}^{σ} q_{s}^{Δ σ} = ξ_{s}^{σ} - ξ_{0}^{σ} (q_{s}^{Δ} + μ (t) q_{s}^{Δ Δ}) . \end{matrix}$ $$\begin{array}{} \displaystyle \xi^{\sigma}_{s}-\xi^{\sigma}_{0}q^{\Delta\sigma}_{s}=\xi^{\sigma}_{s}-\xi^{\sigma}_{0}(q^{\Delta}_{s}+\mu(t)q^{\Delta\Delta}_{s}). \end{array}$$(38)

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

$\begin{matrix} I = \frac{\partial L}{\partial q_{s}^{Δ}} + (L - \frac{\partial L}{\partial q_{s}^{Δ}} q_{s}^{Δ} - \frac{\partial L}{\partial t} μ (t)) ξ_{0}^{σ} + G . \end{matrix}$ $$\begin{array}{} \displaystyle I=\frac{\partial L}{\partial q^{\Delta}_{s}}+\left(L-\frac{\partial L}{\partial q^{\Delta}_{s}} q^{\Delta}_{s}- \frac{\partial L}{\partial t}\mu(t)\right)\xi^{\sigma}_{0}+G. \end{array}$$(39)

Proof

It proves that Eq. (39) is equivalent to the proof of I = const, takeing the derivative of I with respect to t, then

$\begin{matrix} \frac{Δ}{Δ t} I & = \frac{Δ}{Δ t} (\frac{\partial L}{\partial q_{s}^{Δ}} + (L - \frac{\partial L}{\partial q_{s}^{Δ}} q_{s}^{Δ} - \frac{\partial L}{\partial t} μ (t)) ξ_{0}^{σ} + G) \\ = \frac{Δ}{Δ t} \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} + \frac{Δ}{Δ t} (L - \frac{\partial L}{\partial q_{s}^{Δ}} q_{s}^{Δ} - \frac{\partial L}{\partial t} μ (t)) ξ_{0}^{σ} \\ + (L - \frac{\partial L}{\partial q_{s}^{Δ}} q_{s}^{Δ} - \frac{\partial L}{\partial t} μ (t)) ξ_{0}^{σ} + \frac{Δ}{Δ t} G \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\Delta}{\Delta t}I \!\!\!\!\!\!\!\!\!\!&&\displaystyle=\frac{\Delta}{\Delta t}\left[\frac{\partial L}{\partial q^{\Delta}_{s}}+\left(L-\frac{\partial L}{\partial q^{\Delta}_{s}} q^{\Delta}_{s}- \frac{\partial L}{\partial t}\mu(t)\right)\xi^{\sigma}_{0}+G\right]\\ &&\displaystyle=\frac{\Delta}{\Delta t}\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\sigma}_{s}+\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}+\frac{\Delta}{\Delta t}\left[ L-\frac{\partial L}{\partial q^{\Delta}_{s}} q^{\Delta}_{s}- \frac{\partial L}{\partial t}\mu(t)\right]\xi^{\sigma}_{0}\\ &&\displaystyle+\left(L-\frac{\partial L}{\partial q^{\Delta}_{s}} q^{\Delta}_{s}- \frac{\partial L}{\partial t}\mu(t)\right)\xi^{\sigma}_{0}+\frac{\Delta}{\Delta t}G \end{array}$$(40)

From Eq. (22) and (33), we can obtain

$\begin{matrix} \frac{Δ}{Δ t} I & = (\frac{\partial L}{\partial q_{s}^{σ}} + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s})) ξ_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} \\ + (\frac{\partial L}{\partial t} - (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) q_{s}^{σ Δ}) ξ_{0}^{σ} + L ξ_{0}^{Δ} \\ - \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} q_{s}^{Δ} - \frac{\partial L}{\partial t} μ (t) ξ_{0}^{Δ} + \frac{Δ}{Δ t} G \\ = (\frac{\partial L}{\partial q_{s}^{σ}} + (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s})) ξ_{s}^{σ} \\ + \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} + (\frac{\partial L}{\partial t} - (Q_{s}^{″} + Q_{s}^{ω} + Γ_{s} + Λ_{s}) q_{s}^{σ Δ}) (ξ_{0}^{σ} + μ (t) ξ_{0}^{Δ}) \\ + L ξ_{0}^{Δ} - \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} q_{s}^{Δ} - \frac{\partial L}{\partial t} μ (t) ξ_{0}^{Δ} + \frac{Δ}{Δ t} G \\ = \frac{\partial L}{\partial t} ξ_{0} + \frac{\partial L}{\partial q_{s}^{σ}} ξ_{s}^{σ} + \frac{\partial L}{\partial q_{s}^{σ}} ξ_{s}^{σ} + L ξ_{0}^{Δ} - \frac{\partial L}{\partial q_{s}^{Δ}} ξ_{s}^{Δ} q_{s}^{Δ} + (Q_{s}^{″} \\ + Q_{s}^{ω} + Γ_{s} + Λ_{s}) (ξ_{s}^{σ} - ξ_{0}^{σ} q_{s}^{σ Δ}) + \frac{Δ}{Δ t} G \\ = 0 \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\Delta}{\Delta t}I \!\!\!\!\!\!\!\!\!&&\displaystyle=\left( \frac{\partial L}{\partial q^{\sigma}_{s}}+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s}) \right)\xi^{\sigma}_{s}+ \frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}\\ \!\!\!\!\!\!\!\!\!&&\displaystyle+\left( \frac{\partial L}{\partial t}-(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})q^{\sigma\Delta}_{s} \right)\xi^{\sigma}_{0}+L\xi^{\Delta}_{0}\\ \!\!\!\!\!\!\!\!\!&&\displaystyle- \frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}q^{\Delta}_{s}-\frac{\partial L}{\partial t}\mu(t)\xi^{\Delta}_{0}+\frac{\Delta}{\Delta t}G\\ \!\!\!\!\!\!\!\!\!&&\displaystyle=\left( \frac{\partial L}{\partial q^{\sigma}_{s}}+(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s}) \right)\xi^{\sigma}_{s}\\ \!\!\!\!\!\!\!\!\!&&\displaystyle+\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}+\left( \frac{\partial L}{\partial t}-(Q''_{s}+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})q^{\sigma\Delta}_{s} \right)(\xi^{\sigma}_{0}+\mu(t)\xi^{\Delta}_{0})\\ \!\!\!\!\!\!\!\!\!&&\displaystyle+L\xi^{\Delta}_{0}- \frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}q^{\Delta}_{s}-\frac{\partial L}{\partial t}\mu(t)\xi^{\Delta}_{0}+\frac{\Delta}{\Delta t}G\\ \!\!\!\!\!\!\!\!\!&&\displaystyle=\frac{\partial L}{\partial t}\xi_{0}+\frac{\partial L}{\partial q^{\sigma}_{s}}\xi^{\sigma}_{s}+\frac{\partial L}{\partial q^{\sigma}_{s}}\xi^{\sigma}_{s}+L\xi^{\Delta}_{0}-\frac{\partial L}{\partial q^{\Delta}_{s}}\xi^{\Delta}_{s}q^{\Delta}_{s}+(Q''_{s}\\ \!\!\!\!\!\!\!\!\!&&\displaystyle+Q^{\omega}_{s}+\Gamma_{s}+\Lambda_{s})(\xi^{\sigma}_{s}-\xi^{\sigma}_{0}q^{\sigma\Delta}_{s})+\frac{\Delta}{\Delta t}G\\ \!\!\!\!\!\!\!\!\!&&\displaystyle=0 \end{array}$$(41)

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:

$\begin{matrix} T = 2^{n} : n \in N \cup 0 \end{matrix}$ $$\begin{array}{} \displaystyle T={2^{n}:n\in N\cup {0}} \end{array}$$

Suppose lagrangian equation of the system is:

$\begin{matrix} L = \frac{1}{2} m ((q_{1}^{Δ})^{2} + (q_{2}^{Δ})^{2} + (q_{3}^{Δ})^{2}) + \frac{1}{2} m ω^{2} ((q_{1}^{σ})^{2} + (q_{2}^{σ})^{2}) \\ - \frac{1}{2} k ((q_{1}^{σ})^{2} + (q_{2}^{σ})^{2} + (q_{3}^{σ})^{2}) . \end{matrix}$ $$\begin{array}{} \displaystyle L=\frac{1}{2}m\left((q^{\Delta}_{1})^{2}+(q^{\Delta}_{2})^{2}+(q^{\Delta}_{3})^{2}\right)+\frac{1}{2}m\omega^{2}\left((q^{\sigma}_{1})^{2}+(q^{\sigma}_{2})^{2}\right) \\ \displaystyle\quad-\frac{1}{2}k\left((q^{\sigma}_{1})^{2}+(q^{\sigma}_{2})^{2}+(q^{\sigma}_{3})^{2}\right). \end{array}$$(42)

The generalized rotary inertia force and nonconservative force are respectively

$\begin{matrix} Q_{s}^{″} = 0, Q_{s}^{ω} = 0 (s = 1, 2, 3) . \end{matrix}$ $$\begin{array}{} \displaystyle Q''_{s}=0,Q^{\omega}_{s}=0(s=1,2,3). \end{array}$$

The generalized gyroscopic force are

$\begin{matrix} Γ_{1} = 2 m ω q_{2}^{Δ}, Γ_{2} = - 2 m ω q_{1}^{Δ}, Γ_{3} = 0. \end{matrix}$ $$\begin{array}{} \displaystyle \Gamma_{1}=2m\omega q^{\Delta}_{2}, \Gamma_{2}=-2m\omega q^{\Delta}_{1}, \Gamma_{3}=0. \end{array}$$

The Chetaev constraint is

$\begin{matrix} f = q_{2}^{Δ} - t q_{1}^{Δ} = 0 \end{matrix}$ $$\begin{array}{} \displaystyle f=q^{\Delta}_{2}-tq^{\Delta}_{1}=0 \end{array}$$(43)

From Eq. (22), we can obtain

$\begin{matrix} m q_{1}^{Δ Δ} = - k q_{1}^{σ} + m ω^{2} q_{1}^{σ} + 2 m ω q_{2}^{Δ} - λ t, \\ m q_{2}^{Δ Δ} = - k q_{2}^{σ} + m ω^{2} q_{2}^{σ} + 2 m ω q_{1}^{Δ} + λ, \\ m q_{3}^{Δ Δ} = - k q_{3}^{σ} . \end{matrix}$ $$\begin{array}{} \displaystyle mq^{\Delta\Delta}_{1}=-kq^{\sigma}_{1}+m\omega^{2}q^{\sigma}_{1}+2m\omega q^{\Delta}_{2}-\lambda t,\\ \displaystyle \,mq^{\Delta\Delta}_{2}=-kq^{\sigma}_{2}+m\omega^{2}q^{\sigma}_{2}+2m\omega q^{\Delta}_{1}+\lambda ,\\ \displaystyle\qquad\qquad\quad\, mq^{\Delta\Delta}_{3}=-kq^{\sigma}_{3}. \end{array}$$(44)

Using Eq. (42) and Eq. (43), we have

$\begin{matrix} λ = 2 m ω q_{1}^{Δ} + \frac{m q_{1}^{Δ} + (k - m ω^{2}) (q_{2}^{σ} - q_{1}^{σ} t)}{1 + t^{2}} . \end{matrix}$ $$\begin{array}{} \displaystyle \lambda=2m\omega q^{\Delta}_{1}+\frac{mq^{\Delta}_{1}+(k-m\omega^{2})(q^{\sigma}_{2}-q^{\sigma}_{1}t)}{1+t^{2}}. \end{array}$$(45)

The differential equation of the relative motion systems

$\begin{matrix} m q_{1}^{Δ Δ} = - (k - m ω^{2}) \frac{q_{1}^{σ} + t q_{2}^{σ}}{1 + t^{2}} - \frac{m q_{1}^{Δ} t}{1 + t^{2}}, \\ m q_{2}^{Δ Δ} = - (k - m ω^{2}) \frac{q_{1}^{σ} t + t^{2} q_{2}^{σ}}{1 + t^{2}} + \frac{m q_{1}^{Δ}}{1 + t^{2}}, \\ m q_{3}^{Δ Δ} = - k q_{3}^{σ} . \end{matrix}$ $$\begin{array}{} \displaystyle\, mq^{\Delta\Delta}_{1}=-(k-m\omega^{2})\frac{q^{\sigma}_{1}+tq^{\sigma}_{2}}{1+t^{2}}-\frac{mq^{\Delta}_{1}t}{1+t^{2}},\\\displaystyle mq^{\Delta\Delta}_{2}=-(k-m\omega^{2})\frac{q^{\sigma}_{1}t+t^{2}q^{\sigma}_{2}}{1+t^{2}}+\frac{mq^{\Delta}_{1}}{1+t^{2}},\\\displaystyle\qquad\qquad\quad mq^{\Delta\Delta}_{3}=-kq^{\sigma}_{3}. \end{array}$$(46)

From the Theorem 3, we obtain

$\begin{matrix} ξ_{1}^{σ} (m ω^{2} q_{1}^{σ} - k q_{1}^{σ}) + ξ_{2}^{σ} (m ω^{2} q_{2}^{σ} - k q_{2}^{σ}) - ξ_{3}^{σ} k q_{3}^{σ} + m q_{1}^{Δ} (ξ_{1}^{Δ} - ξ_{0}^{Δ} q_{1}^{Δ}) \\ + m q_{2}^{Δ} (ξ_{2}^{Δ} - ξ_{0}^{Δ} q_{2}^{Δ}) + m q_{3}^{Δ} (ξ_{1}^{Δ} - ξ_{0}^{Δ} q_{3}^{Δ}) + (\frac{1}{2} m ((q_{1}^{Δ})^{2} + (q_{2}^{Δ})^{2} + (q_{3}^{Δ})^{2})) \\ (+ \frac{1}{2} m ω^{2} ((q_{1}^{σ})^{2} + (q_{2}^{σ})^{2}) - \frac{1}{2} k ((q_{1}^{σ})^{2} + (q_{2}^{σ})^{2} + (q_{3}^{σ})^{2})) ξ_{0}^{Δ} \\ + (2 m ω q_{2}^{Δ} - 2 m ω q_{1}^{Δ} t - \frac{m q_{1}^{Δ} + (k - m ω^{2}) (q_{2}^{σ} - q_{1}^{σ} t)}{1 + t^{2}} t) \times (ξ_{1}^{σ} - ξ_{0}^{σ} q_{1}^{σ Δ}) \\ + (2 m ω q_{1}^{Δ} - 2 m ω q_{1}^{Δ} + \frac{m q_{1}^{Δ} + (k - m ω^{2}) (q_{2}^{σ} - q_{1}^{σ} t)}{1 + t^{2}}) \times (ξ_{2}^{σ} - ξ_{0}^{σ} q_{2}^{σ Δ}) \\ = - \frac{Δ}{Δ t} G . \end{matrix}$ $$\begin{array}{} \displaystyle \xi^{\sigma}_{1}(m\omega^{2}q^{\sigma}_{1}-kq^{\sigma}_{1})+\xi^{\sigma}_{2}(m\omega^{2}q^{\sigma}_{2}-kq^{\sigma}_{2})-\xi^{\sigma}_{3}kq^{\sigma}_{3} +mq^{\Delta}_{1}(\xi^{\Delta}_{1}-\xi^{\Delta}_{0}q^{\Delta}_{1})\\ \displaystyle+mq^{\Delta}_{2}(\xi^{\Delta}_{2}-\xi^{\Delta}_{0}q^{\Delta}_{2}) +mq^{\Delta}_{3}(\xi^{\Delta}_{1}-\xi^{\Delta}_{0}q^{\Delta}_{3})+\left[\frac{1}{2}m\left((q^{\Delta}_{1})^{2}+(q^{\Delta}_{2})^{2}+(q^{\Delta}_{3})^{2}\right)\right.\\ \displaystyle\left.+\frac{1}{2}m\omega^{2}\left((q^{\sigma}_{1})^{2}+(q^{\sigma}_{2})^{2}\right) -\frac{1}{2}k\left((q^{\sigma}_{1})^{2} +(q^{\sigma}_{2})^{2}+(q^{\sigma}_{3})^{2}\right)\right]\xi^{\Delta}_{0}\\ \displaystyle+\left[2m\omega q^{\Delta}_{2}-2m\omega q^{\Delta}_{1}t-\frac{mq^{\Delta}_{1}+(k-m\omega^{2})(q^{\sigma}_{2}-q^{\sigma}_{1}t)}{1+t^{2}}t \right]\times(\xi^{\sigma}_{1}-\xi^{\sigma}_{0}q^{\sigma\Delta}_{1})\\ \displaystyle+\left[2m\omega q^{\Delta}_{1}-2m\omega q^{\Delta}_{1}+\frac{mq^{\Delta}_{1}+(k-m\omega^{2})(q^{\sigma}_{2}-q^{\sigma}_{1}t)}{1+t^{2}} \right]\times(\xi^{\sigma}_{2}-\xi^{\sigma}_{0}q^{\sigma\Delta}_{2})\\ \displaystyle=-\frac{\Delta}{\Delta t}G. \end{array}$$(47)

Choosing the infinitesimal generators as:

$\begin{matrix} ξ_{0} = ξ_{1} = ξ_{2} = 0, ξ_{3} = q_{3}^{Δ} . \end{matrix}$ $$\begin{array}{} \displaystyle \xi_{0}=\xi_{1}=\xi_{2}=0,\xi_{3}=q^{\Delta}_{3}. \end{array}$$(48)

Using Eqs. (47-48), we have

$\begin{matrix} G = \frac{k}{2} q_{3}^{σ} - \frac{1}{2} m (q_{3}^{Δ})^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle G=\frac{k}{2}q^{\sigma}_{3}-\frac{1}{2}m(q^{\Delta}_{3})^{2}. \end{array}$$

According to Theorem 4, the system has the Noether conserved quantity on time scale as

$\begin{matrix} I = \frac{1}{2} m (q_{3}^{Δ})^{2} + \frac{1}{2} k (q_{3}^{σ})^{2} = c o n s t . \end{matrix}$ $$\begin{array}{} \displaystyle I=\frac{1}{2}m(q^{\Delta}_{3})^{2}+\frac{1}{2}k(q^{\sigma}_{3})^{2}=const. \end{array}$$(49)

Conclusion

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.

eISSN:: 2444-8656
Language:: English

Publication timeframe:: 2 times per year
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

Journal RSS Feed

Noether’s theorems for the relative motion systems on time scales

Published Online: Dec 01, 2018

Page range: 513 - 526

Received: Aug 26, 2018

Accepted: Nov 26, 2018

DOI: https://doi.org/10.2478/AMNS.2018.2.00040

Keywordstime scale, Noether theorem, the relative motion systems

© 2018 Sheng-nan Gong and Jing-li Fu, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
time scale, Noether theorem, the relative motion systems